While at one point, Hezbollah (In Arabic, Hizb Allah means "Party of God") was the best group at killing Americans, Hezbollah focuses a lot of attention on fighting Israel. Indeed, to act as a resistance to Israeli occupation is an outlining factor into why they were formed. In recent years, however, their attention has faded away from Israel and to helping Bashar al-Assad stay in power in Syria, as well as supporting Iran's shadow war with the West. Having said that, the Party of God could begin to shift back to fighting Israel.
Hezbollah was formed in the early 1980's as a result of Iran wanting a Lebanese-proxy against Israel (Levitt). As I said before, Hezbollah was formed, in part, to be the resistance against Israeli occupation following the 1982 Israeli invasion of Lebanon. Iran, for their part, made sure each new recruit attended an Iranian Revolutionary Guards Corp-Quds Force (or simply, Quds Force) training camp in the Bekaa Valley (Levitt). Hezbollah is also known to vehemently support an Iranian ideology called velayat-e faqih (guardian ship of the jurist). This ideology essentially says that a Shia cleric can also serve as the head of government. It is thought that Iran still gives Hezbollah around $200 million to fund their terrorist activities, their political and social activities, as well as their media wing (Levitt).
Since the 1980's, however, they have developed an extensive network of global connections and ties to various militant groups and even Mexican drug cartels. Utilizing these connections allows for Hezbollah to support Iranian activities around the globe. For instance, it is thought Hezbollah acted upon the behest of Iran when they launched the Burgas terrorist attack in Bulgaria in 2012. Even further, several Hezbollah operatives were arrested in Baku, Azerbaijan in 2006 and 2008 and charged with plotting against Israeli and American interests there (Levitt).
These international connections are thought to be as a result of both Iranian influence of the group and the ambitions of the leader of their international wing, Imad Mughniyah. Mughniyah, born in Lebanon, was not only in charge of their international wing, but also their military and intelligence wings (Levitt). Mughniyah was thought to be behind some of the groups most spectacular and devastating bombings and hijacking in the 1980's. In fact, the CIA has accused him of killing more Americans than any other militant before 9/11. He was killed in 2008 in Damascus, Syria by an assumed Mossad assassination.
Their activities now:
"Over the course of the always intimate relationship between Iran and Hezbollah, the head of the Quds Force or other senior Iranian leaders might have told Hezbollah to "jump" and the response would have been, "How high?" (Levitt). Based on what this passage tells us, it is easy to surmise that Hezbollah does the bidding of whatever Iran wishes. This is why we saw Hezbollah operatives perpetrating a terrorist attack in Bulgaria, why several operatives were arrested in Azerbaijan plotting an attack, and why several members were arrested in Thailand and Cyprus for the same thing (Levitt). It should come as no surprise as to why Hezbollah is currently fighting on the side of the regime in Syria. The Assad regime is a key partner to both Iran and Hezbollah, and Iran needs Assad to stay in power. Hezbollah seems more than happy to intervene in Syria at the behest of their masters in Iran.
Even further, Iran asked the same of Hezbollah during the US-led War in Iraq. During the fiercest time of sectarian killings in Iraq (2006-2007), a Hezbollah operative by the name of Ali Moussa Daqduq was charged by Iran to be their liaison in the coordination of the training of several Shia militant groups (Levitt). But not only that, Hezbollah operated several training camps for several Iraqi Shia militant groups--these groups were dubbed the "Special Groups" by the United States. This only further shows their commitment to Iran.
Returning the Focus to Israel:
Despite being a force to be reckoned with in the battlefields of Syria, Hezbollah's activities in the country are not coming without a price. Indeed, several car bombings attributed to Sunni al-Qaeda organizations, Abdullah Azzam Brigades and Jabhat al-Nusra in Lebanon have targeted both Hezbollah and Iranian assets. As a result, Hezbollah's popularity has been seemingly declining among the Lebanese people. The people do not want to upset the fragile balance between Sunni's and Shi'ites in Lebanon; this is something Hezbollah is seen as toying with in their intervention in Syria. As more bombings, fighting and chaos occurs in Lebanon due to spillover in Syria, Hezbollah will probably continue to lose support. In the event that Hezbollah feels like they need to regain support from the Lebanese Shia community, their crosshairs could turn back to Israel. This could be already happening with a recent bombing on Israeli military outpost by Hezbollah. Hezbollah will have to prove to the people of Lebanon, whom they have been supporting since the 80's with various social services, that their focus is still on Israel.
Returning to Iraq?
Being that Hezbollah has operated in Iraq before, it stands to reason that Hezbollah could begin to be involved yet again in the wake of the Islamic State of Iraq and ash-Sham's blitzkrieg in Iraq. The ISIS blitz has since set off a fervor of sectarian passion among both Sunni's and Shi'ites. In fact, the head Shia cleric in Iraq has issued a call to arms and Moqtada al-Sadr's infamous Mahdi Army has thought to have been mobilized again. If push comes to shove in Iraq and we start to see another sectarian civil war in Iraq, I would not doubt Hezbollah will again be operating in Iraq at the behest of Iran. Now, being that they want to rollback their negative image in Lebanon, we may not see the same levels of intervention as in Syria or previously in Iraq. I believe training of troops would be the most obvious contribution to the crisis by Hezbollah. I also do not rule out the possibility of this already happening being that the Quds Force is thought to be helping in the fight against ISIS.
By going to Syria at the behest of their patron Iran to defend their mutual ally Bashar, it removes any doubt that they most certainly are a global organization doing the bidding of Iran--leaving the citizens of Lebanon, whom they are supposed to protect, susceptible to more violence. This violence will be from Sunni groups, like the AQ-linked Abdullah Azzam Brigades or even al-Nusra, fighting between Sunnis and Shi'ites in the Bekaa because of Syria, or even the Lebanese state. This will most definitely upset the balance Hezbollah has tried to maintain.
Moreover, I see Hezbollah trying to attack Israel to be even more likely the longer they are in Syria. This has nothing to do with expanding that war; rather, its about shifting the focus, or appearing to shift it, back to Israel, the original enemy. Inadvertently in doing so, however, this will probably result in the expansion of the war in Syria. But the thinking in doing so, in Hezbollah's eyes, would be to show the Shi'ites of Lebanon that they are still the vanguard against Israel and maintain the power and influence they wield in Lebanon. Hezbollah also stands a good chance of getting involved in Iraq the more Iran gets involved in Iraq.
Hezbollah: The Global Footprint of Lebanon's Party of God by Matthew Levitt.