Not sure...
Queen to F1 - he has to step back his King to H4
Queen to H1 - he'll bring the Rook to H3 (if he doesn't and moves the King to G5, Mate with Pawn F7-F6 (or Queen to H5) - that would be Mate in 3, so not what you were looking for)
Queen to E1 - he'll bring the Rook to G3 (if he doesn't and moves the King to G5, Mate with Pawn F7-F6 - that would be Mate in 4, but not the best scenario for Whites)
Queen to G3 - he'll move the King to G5
Mate with Pawn F7-F6
That's 5. Can't see an ideal 4, what am I missing?