(Gulf Cooperation Council is imprecise shorthand for Saudi Arabia and aligned states such as Bahrain and Qatar)
I don't think there is GCC political will to fight ISIS. The anti-ISIS maneuvers have been anemic at best, where they have even existed. This was true before the Houthi coup which tried to change Yemen. Partially this is because GCC want to see Assad ousted, but they also want a resurgent Shia Iraq to be destabilized. Remember that Turkey has bought ISIS oil. And at least in the past, Gulf states helped fund ISIS.
ISIS is on some levels a continuation of the Iraqi regime that was trying to destroy Iran to solidify Saudi/Gulf dominance in the region (Iran/Iraq war). Remember there is some continuity from Baathist Iraqi elites and IS leadership today. The Houthi are a real threat to GCC states, but ISIS is at least composed partially of a GCC/Saudi ally - and remember the Saudis opposed the invasion of Iraq that toppled that ally.
I think quite a bit more is going on "behind the scenes" in the form of diplomacy (or at least calculation) by Gulf states toward ISIS. Even if there is no trust or communication line, there is an understanding - ISIS's existential threats are Shia Iraq and Assad in Syria - both allied with Iran. These are all Saudi enemies. Saudi Arabia is quite happy to have a Sunni thorn in their sides.
Moreover, ISIS is ruthlessly tactical (see leaked documents which show how ISIS used covert agents and power brokers to take over its cities). Tactical states in the region understand that there is a brutally rational core to ISIS which could possibly be negotiated with, or manipulated.