Let $G$ be a group and $H$ solvable subgroup of index smaller or equal to $4$. Show $G$ is a solvable group.
I was thinking about letting $G$ act on $G/H$ and then use a lemma that if $\ker\phi $ and ${\rm im}\, \phi$ are solvable then $G$ is solvable. But I got stuck
Proof (sketch) Assume $|G:H| \leq 4$. Let $G$ act by left multiplication on the left cosets of $H$. The kernel of this action is ${\rm core}_G(H)=\bigcap_{g \in G}H^g$, the largest normal subgroup of $G$ contained in $H$. Hence $G/{\rm core}_G(H)$ can homomorphically be embedded in $S_{|G:H|}$. Since $S_4$ is solvable and ${\rm core}_G(H)$ as subgroup of $H$ is solvable, it follows that $G$ is solvable.