Normality of subgroups and Sylow groups under homomorphisms...

Normality of subgroups and Sylow groups under homomorphisms...

Prove or disprove: if $f:G \rightarrow H$ is a surjective group
homomorphism, then
1) $f(A)$ normal implies $A$ normal.
2) $S$ is a p-Sylow subgroup of $G$ implies $f(S)$ is a Sylow-p subgroup
of $H$.
I believe 1 is false since I believe we need injectivity, but I cannot
think of a counterexample...
I believe 2 is true but do not know how to show it...