In my research I need to compute the group homology of the dicyclic group Dic3, which is a semi-direct product of $\mathbb{Z}/3\mathbb{Z}$ and $\mathbb{Z}/4\mathbb{Z}$, and let's denote it by $G$, we have a short exact sequence,
\begin{equation}
0 \rightarrow \mathbb{Z}/3\mathbb{Z} \rightarrow G \rightarrow \mathbb{Z}/4\mathbb{Z} \rightarrow 0
\end{equation}
My idea is to use Lyndon–Hochschild–Serre spectral sequence
\begin{equation}
H_p(\mathbb{Z}/4\mathbb{Z}, H_q(\mathbb{Z}/3\mathbb{Z},\mathbb{Z})) \Rightarrow H_{p+q}(G,\mathbb{Z})
\end{equation}
We already know that
\begin{equation}
H_2(\mathbb{Z}/4\mathbb{Z}, \mathbb{Z})=H_2(\mathbb{Z}/3\mathbb{Z}, \mathbb{Z})=0
\end{equation}
so we deduce that
\begin{equation}
H_2(G,\mathbb{Z})=H_1(\mathbb{Z}/4\mathbb{Z}, H_1(\mathbb{Z}/3\mathbb{Z},\mathbb{Z}))
\end{equation}
We also know
\begin{equation}
H_1(\mathbb{Z}/3\mathbb{Z},\mathbb{Z})=\mathbb{Z}/3\mathbb{Z}
\end{equation}
**Question 1**: Does $\mathbb{Z}/4\mathbb{Z}$ acts trivially on $H_1(\mathbb{Z}/3\mathbb{Z},\mathbb{Z})$?

If so, we could conclude that $H_2(G,\mathbb{Z})=0$.

**Question 2**: If the action of $\mathbb{Z}/4\mathbb{Z}$ on $H_1(\mathbb{Z}/3\mathbb{Z},\mathbb{Z})$ is not trivial, do we still have
$$H_1(\mathbb{Z}/4\mathbb{Z}, H_1(\mathbb{Z}/3\mathbb{Z},\mathbb{Z})) =0 $$