Prove that KS/K is a p-Sylow subgroup of G/K.

Proof/

So we know that K is a subgroup of KS and KS is a subgroup of G, so by the correspondence theroem KS/K is a subgroup of G/K. Since |KS/K|=|S|/|K intersect S|, then |KS/K|= p (where p is a prime).

Want to show that KS/K is a miaximal p-subgroup of G/K.

So it suffices to show that [G/K:KS/K] is relatively prime to p. From here is where I'am having trouble I have been trying to use the facts about S being a p-sylow subgroup of G, to get that S/K is a p-sylow subgroup of G/K to try to get my conclusion,but I have been getting no where doing this. Any suggestions would be a life saver.