Group action and semidirect product Let $m,n \in \mathbb N$, $k$ a field, $X=(k^{n\times m},+)$, and we consider the groups $GL(n,k)$ and $GL(m,k)$. Let $K:=GL(n,k) \times GL(m,k)$. We define
\begin{align*}
K\times X &\to X\\
((A,B),M) &\mapsto AMB^{-1}
\end{align*}
Note that this defines an action of $K$ on $X$. With this action we can consider $X \rtimes K$. Let
$$
G=\left\{\begin{pmatrix}
        A & M \\
        0 & B \\
        \end{pmatrix} :  A \in GL(n,k), B \in GL(m,k), M \in X \right\} \, .
$$
Show that $G \cong X \rtimes K$.
I've tried to define the following map
\begin{align*}
\psi:G &\to X \rtimes K\\
\begin{pmatrix}
        A & M \\
        0 & B \\
        \end{pmatrix} &\mapsto (M,(A,B))
\end{align*}
but this is not a morphism since $$\begin{pmatrix}
        A & M \\
        0 & B \\
        \end{pmatrix}\begin{pmatrix}
        C & M' \\
        0 & D \\
        \end{pmatrix}=\begin{pmatrix}
        AC & AM'+MD \\
        0 & BD \\
        \end{pmatrix},$$ so $$\psi\left(\begin{pmatrix}
        A & M \\
        0 & B \\
        \end{pmatrix}\begin{pmatrix}
        C & M' \\
        0 & D \\
        \end{pmatrix}\right)=(AM'+MD,(AC,BD)),$$ and  $$\psi\left(\begin{pmatrix}
        A & M \\
        0 & B \\
        \end{pmatrix}\right)\psi\left(\begin{pmatrix}
        C & M' \\
        0 & D \\
        \end{pmatrix}\right)=(M,(A,B)).(M',(C,D))=(M+AM'B^{-1},(AC,BD))$$
Could someone suggest me an appropriate isomorphism between these two groups? Thanks in advance.

First we identify the subgroups of $G$ corresponding to $X$ and $K$:
\begin{align*}
X &\cong  \left\{
\begin{pmatrix}
1 & M\\
0 & 1
\end{pmatrix}
: M \in X \right\} =:\tilde{X}\\
\quad
K &\cong \left\{
\begin{pmatrix}
A & 0\\
0 & B
\end{pmatrix}
: A \in GL_n(k), B \in GL_m(k)\right\} =: \tilde{K}
\end{align*}
where $1$ is the identity matrix of the appropriate size.
Note that
\begin{align*}
\begin{pmatrix}
A & M\\
0 & B
\end{pmatrix}
=
\begin{pmatrix}
1 & M B^{-1}\\
0 & 1
\end{pmatrix}
\begin{pmatrix}
A & 0\\
0 & B
\end{pmatrix} \in \tilde{X} \tilde{K} \, .
\end{align*}
This suggests the map
\begin{align*}
\varphi : G & \to X \rtimes K\\
\begin{pmatrix}
A & M\\
0 & B
\end{pmatrix} & \mapsto (MB^{-1}, (A,B)) \, .
\end{align*}
Can you show that this map is a homomorphism?
