The U.S. may not want "boots on the ground" in Iraq, but Iran sure does. In Iraqi Kurdistan, Iran's military involvement in the Kurdish governorates of northern Iraq is multi-varied and on the increase.
Kurdish Rudaw T.V. has reported on Iran's support for Kurdistan's Peshmerga (military) campaign to regain villages lost to Islamic State [IS] jihadists this past summer. Rudaw T.V. even discussed the public visits of Iran's Major General Qasem Soleimani, commander of the Revolutionary Guard's Qods Force, to the Peshmerga front line against the IS.
According to a former Iraqi General of Military Intelligence, who paid a mid-November visit to Kurdistan, the big winner politically in the multilateral effort to roll back the IS's territorial gains is Iran. General Saad al-Obaidi commented that without the presence of several pro-Iran Shia militias and Iranian artillery support, allied bombing raids against IS targets would have been for naught.
According to Saad, the most effective pro-Iran militia is the storied Badr Corps, which had long sought to topple the Saddam Hussein regime. Asaeeb Ahl al-Haq (House of the Truth) fighters are another Shia group closely allied with the Islamic Republic of Iran. The Iraqi Hezbollah is still another pro-Tehran militia. Some of the warriors from this militia originally received their training in camps in Iran's Ahvaz Province. These facilities were utilized to train militias that targeted American troops in Iraq after the overthrow of Saddam Hussein.
Specific operations where the presence of volunteers from the Iranian military's Islamic Revolutionary Guard Corps [IRGC] helped Kurdish Peshmerga troops to regain lost territory took place during late summer battles in Diyala Province.  Iranian pilots, probably with their F-4 Phantom fighter aircraft, also have flown a few sorties against Islamic State positions.
Tehran even has established an unofficial "no go zone" across from its northwestern border inside Iraqi Kurdistan's Sulaimaniyah Governorate. Iran's most invidious influence, however, is possibly the widespread invisible presence of agents from its Ministry of Intelligence [MOIS].
U.S. and Western policymakers might want to give serious thought to what degree they wish to diminish the IS; too great a setback to these Sunni jihadists will only serve the interest of pro-Iranian Shia militias.
Dr. Lawrence A. Franklin served on active duty with the U.S. Army and as a Colonel in the Air Force Reserve, where he served as a Military Attaché to Israel.
 Kurdistan: Iraq News in Brief, 26 November 2014. Rudaw TV: "Shiite Militia Commander: No Kurdish Force Took Part in Saadiya Liberation".
 General Saad al-Obaidi, (ret.) Asharq Centre for Research and Consulting, Interview, 24 November 2014.
 Military Leadership Monitor. November 2014 Issue. "A Portrait of Iran's Legendary IRGC Commander Qasem Soleimani".