Let $G$ be a finite group with properties listed in the following.
(1) $G/G''\cong A_4$.
(2) $G/K\cong SL(2,3)$ where $K=F(G)$ is isomorphic to an extraspecial $3$-group of order $3^7$, $K/Z(K)$ is a $G$-chief factor.
(3) Let $P\in Syl_3(G)$. Then $P=KD$ where $D\cong C_{9}$ and $Z(P)=Z(K)=Z(G)$.
(4) All elements of order $3$ lie in $G'$.
My question is: Does the group that I described here exist?
The group I described here is too big to construct by using GAP (by myself).
Assume that group $G$ that I described exists. Then $G/K$ acting irreducibly on a $GF(3)$-module $V:=K/Z(G)$. So, by Clifford theory, $V$, as a $GF(3)[G'/K]$-module, equals $W\oplus Wx\oplus Wx^2$, where $W$ has dimension 2 and $o(x)=3$. (Otherwise, $V$ is an irreducible $GF(3)[G'/K]$-module, however, $G'/K\cong Q_8$ only has irreducible $GF(3)[G'/K]$-modules of dimension $1$ or $2$.) To construct $G$, I first try to construct a group $T$ isomorphic to $G/Z(G)$, then I construct a factor group of schur covering group of $T$ with center a cyclic subgroup of order $3$.
Any explanation, references, suggestion and examples are appreciated.