Recently I've came to result that, if $H$ is a simple group, every homomorphism $\theta :K\rightarrow \mathrm{Out}(H)$ determines an unique extension of $H$ by $K$. As an example, I tried to find all extensions of $A_5$ by $C_2$, and would like to know if I got to the right conclusions.
Indeed, we have that $\mathrm{Aut}(A_5)\cong S_5$. Once $A_5$ is simple, its center is trivial- and, as a consequence, $\mathrm{In}(A_5)\cong A_5$ and $\mathrm{Out}(A_5)\cong C_2$. Therefore, the only extensions of $A_5$ by $C_2$ are pullbacks of the form $$C_2 \times _{C_2} S_5.$$
These pullbacks are determined by the homomorphisms $\theta :C_2\rightarrow C_2$, of which there are two: the trivial and the constant maps. As a consequence, we have two- and only two- extensions of $A_5$ by $C_2$. Namely, if $\pi$ is the projection of $S_5$ onto $S_5/A_5\cong C_2$, in the trivial case we get $$\{(s,m)|s\in S_5,\, m\in C_2,\, \theta(m)=\pi(s)=0\}\cong A_5 \times C_2,$$
and in the constant case we get $$\{(s,m)|s\in S_5,\, m\in C_2,\, \theta(m)=\pi(s)=m\}=\langle(0,A_5),(1,S_5-A_5)\rangle \cong A_5\rtimes C_2.$$
Is this correct?
Note that your map $\pi$ is more commonly denoted by $\operatorname{sgn}$, as it is the sign map on $S_5$. In the case that $\theta$ is trivial, you get the pullback of groups \begin{eqnarray*} S_5\times_{C_2}C_2 &=&\{(s,m)|\ s\in S_5,\, m\in C_2,\, \theta(m)=\pi(s)\}\\ &=&\{(s,m)|\ s\in S_5,\ m\in C_2,\ \operatorname{sgn}(s)=0\}\\ &=&A_5\times C_2, \end{eqnarray*} which is an equality of sets and hence of groups. Similarly, if $\theta$ is nontrivial then \begin{eqnarray*} S_5\times_{C_2}C_2 &=&\{(s,m)|\ s\in S_5,\, m\in C_2,\, \theta(m)=\pi(s)\}\\ &=&\{(s,m)|\ s\in S_5,\ m\in C_2,\ m=\operatorname{sgn}(s)\}\\ &=&\{(s,\operatorname{sgn}(s)):\ s\in S_5\}, \end{eqnarray*} which is again an equality of sets, and as a group this is clearly isomorphic to $S_5$. Expressing this as $A_5\rtimes C_2$ is correct but not the clearest way of describing the group. And I'm not sure how to interpret $$\langle(0,A_5),(1,S_5-A_5)\rangle,$$ where I assume you switched the order of the factors in the product.