Let $n=3^m$ for some positive integer $m$. Let $G\leq S_n$ be a permutation group on $n$ letters. Denote the largest normal subgroup of $G$ with odd order by $O_{2'}(G)$. My question is the following:

Does there exist $G$ such that $G/O_{2'}(G)\cong A_4$ or $S_4$ for suitable $m$?

Let $n=3^m$ for some positive integer $m$. Let $G\leq S_n$ be a transitive permutation group on $n$ letters. Denote the largest normal subgroup of $G$ with odd order by $O_{2'}(G)$. My question is the following:

Does there exist $G$ such that $G/O_{2'}(G)\cong A_4$ or $S_4$ for suitable $m$?

Let $n=3^m$ for some positive integer m. Let $G\leq S_n$ be a permutation group on n letters. Denote the largest normal subgroup of $G$ with odd order by $O_{2'}(G)$  . My question is the following:

Does there exist $G$ such that $G/O_{2'}(G)\cong A_4$ or $S_4$ for suitable m?

Let $n=3^m$ for some positive integer $m$. Let $G\leq S_n$ be a permutation group on $n$ letters. Denote the largest normal subgroup of $G$ with odd order by $O_{2'}(G)$. My question is the following:

Does there exist $G$ such that $G/O_{2'}(G)\cong A_4$ or $S_4$ for suitable $m$?