Alice's connection to Dave Grohl's 'Foo Fighters' seems to be primarily through guitarist Pat Smear. Smear is apparently a HUGE Cooper fan, who has apparently in the past cited Alice Cooper as the reason he started playing guitar.
Alice has performed with the band a few times. In 2011 he joined the band at their huge Milton Keynes show in the UK to perform 'School's Out' and 'I'm Eighteen' to a huge roar from the crowd. They did the same songs in 2015 when he made a guest appearance at Grohl's big birthday bash at the LA forum, and then for Halloween 2017 Alice performed on 'The Late Show' backed by the Foo Fighters for 'Ballad Of Dwight Fry' complete with straitjacket, and a segment from 'Killer' for the guillotine.
Foo Fighters front man Dave Grohl also appears on the 'Hollywood Vampires' album playing drums on 'One/Jump In The Fire'.
Since then The Foo Fighters have occasionally played other Alice Cooper songs in their shows including most recently 'Under My Wheels' sung by guitarist Chris Shiflett.