Extensions of $A_5$ by $C_2$

Question

I'm attempting the following exercise:

Prove that there are only three finite groups whose composition factors are $A_5$ and $C_2$.

1. The direct product $A_5 \times C_2$.
2. The symmetric group $S_5$.
3. A central extension of $A_5$ by $C_2$ known as the binary icosahedral group.

I've completed parts 1 and 2. For part 3, I'm trying to look at extensions of the form:

$$1 \to C_2 \to G \to A_5 \to 1$$

Is there any straightforward argument to show that there cannot exist a non-central split extension of this form? Furthermore, I would be interested in knowing if there is any neat way to show that the unique non-split central extension of $A_5$ by $C_2$ is isomorphic to $I_{120}$. (@DerekHolt gave some hints in the comments.)