Let's take a trip down memory lane to understand what the USA, UK and others bombing IS in Iraq is really about, starting with a quiz:
What started on December 24th, 1979?
What was Operation Cyclone?
Who did Jama'at al-Tawhid wal-Jihad originally pledge allegiance to, and what did that organization change its name to on October 17th, 2004, and then in 2013/14?
TL:DR; Your answer is that IS are the renamed, rebooted, latest incarnation of the Al Qaida in Iraq franchise. (Same flavour of ideology, still tastes like shit to western tongues.) They got their doctrine from Azzam by way of his sidekick Osama bin Laden, and Azzam got his training and experience in running a big chunk of the insurgency in Afghanistan, with guns and ammo from Pakistani military intelligence (the ISI) bankrolled by the CIA.
It's not hard to follow: the chain of mergers, name-changes, leadership switches, and loyalty goes all the way back to Sayyid Qutb. IS is simply Al Qaida staging a coup in the wake of civil war, and applying Leninist praxis to overthrowing the tenuous grip of Ba'athist nation-building in order to create a revolutionary Caliphate. The only reason our media aren't calling them Al Qaida is because (a) the last administration cried wolf several dozen times too often, and (b) the boil has turned into full-blown necrotising fasciitis and they don't want to scare the horses.
It has not escaped my attention that Al Qaida has effectively grown from a bunch of student radicals sitting around in a basement talking smack about islamist politics, through the same phase of bomb-throwing and bank robberies that Lenin's followers went through in the 1890s and 1900s, to an army with mobile artillery that controls several million people and a bunch of oil wells. There is an ontology to the growth of revolutionary organizations and AQ — or IS as we're now meant to call them — are following the same curve as Mr. Ulianov's vanguard party and drinking society, which snowballed from a bunch of exiles clogging up central European cafes in 1914 to a full-scale government (firing squads, army, secret police and all) by 1918. As with Lenin, idiot western governments who thought they could use the beast for their own ends fed it until it grew big enough to bite the master's hand ... and now we're all supposed to panic and write these same idiots a blank check.
(Parenthetic digression: obviously the CIA didn't directly intend to train and arm these aforementioned dangerous assholes: if they'd known where it would all go they'd have been horrified. But in the 1980s they didn't have a clue about Qutbism, so they handed an open chequebook to ISI, who in turn doled out cash and guns to anyone who would inflict grief on the Soviets in Afghanistan, and perhaps if Zbignew Brzezinski's state department had been slightly more forward-looking and clear eyed they might have pondered where all these angry young men with leet bomb-making skillz were going to go once the Soviets were defeated ... but that's all water under the bridge. Or is it? One thing's certainly true: western governments' track record in picking proxies to fight their wars for them are generally so disastrous that it's almost as if they were looking for rabid dogs. Hmm, time to re-read Chomsky.)
And you know something else? If George W. Bush hadn't had such a raging hard-on for Saddam Hussein, if he hadn't railroaded everyone into invading Iraq, this needn't have happened. Al Qaida have grown into a full-scale scary government-shaped object with a revolutionary ideology because Bush created a power vacuum for them to expand into. (And Obama helped, by not actively propping up the weak Ba'athist regime in Syria — who are bastards, but at least they're not trying to destroy western civilization for a hobby.)
It is to weep. But we've made our bed and now I suppose we must drop bombs on it.