Amount of homomorphisms from $V$ to $S_4$

Question

I am trying to count the amount of homomorphisms from the klein four group to $S_n$, so the homommorphisms $f: V_4\to S_n$.

I think I am almost there, but just wanted to let you guys know my way of reasoning, and if that is correct: the elements of $V_4$ all have order 2 (except the identity). So they all need to go to an element of order 2 in $S_4$, right?

The elements of $S_4$, with order 2, are: (1 2), (1 3), (1 4), (2 3), (2 4), (3 4), (1 2)(3 4), (1 3)(2 4), (1 4)(2 3). So that is 9 elements.

Then I am a bit stuck. Am I at least going in the right direction?

Answer 1

Hint: $V \cong C_2 \times C_2$. Consider the possible homomorphisms $C_2 \to S_4$.