Is the nontrivial semidirect product of $C_3^2$ by $Q$ unique?

Question

Is the nontrivial semidirect product of $C_3^2$ by $Q$ unique?

This is an extension of a homework problem about group representations, but some of the things I can extract from the group characterization (or perhaps, characters) are:

1. The group $G$ is of order $72.$
2. The Sylow $3$-subgroup $H\cong C_3^2$ and is normal.
3. $G/H\cong Q$
4. All elements of $G$ are of order $1, 2, 3,$ or $4$

Answer 1

Between @Shaun and @ArturoMagidin comments, you've answered my question.

Let me summarize: Condition 4 means the semi-direct product corresponds to an embedding from $G/H$ into $\operatorname{Aut}(H) \cong GL_2(3)$. But the second derived subgroup of $GL_2(3)$ is the copy of $Q$ in $GL_2(3)$.

Hence $G \cong \operatorname{Aff}_2(3)''$