In 'A presentation of $PGL(2,p)$ with three defining relations' by E.F.Robertson and P.D.Williams, we can find a presentation of $PGL(2,p)$:

$\langle a,b | a^2 = b^p = (a b^2 a b^r)^2 = (abab^r)^3 = 1 \rangle$, s.t. $r$ is a square-free primitive element of $\mathbb{Z}/p \mathbb{Z}$.

Where we can associate $a$ to $\left( \begin{array}{cc} 0 & -r \\ 1 & 0 \\ \end{array} \right)$ and $b$ to $\left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \\ \end{array} \right)$

It seems it is easy to extend this to $GL(2,\mathbb{Z}/p \mathbb{Z})$ by introducing another generator, $c$, which would be associated to the matrix $\left( \begin{array}{cc} r & 0 \\ 0 & r \\ \end{array} \right)$:

$\langle a,b,c | a^2 = b^p = c^{p-1} = (a b^2 a b^r)^2 = (abab^r)^3 = 1, ac=ca, bc=cb \rangle$

My question is if we can extend this again to $GL(2,\mathbb{Z}/p^n \mathbb{Z})$?