Order of this group Given $k\in{\mathbb{N}}$, we denote $\Gamma _2(p^k)$ the multiplicative group of all matrix $\begin{bmatrix}{a}&{b}\\{c}&{d}\end{bmatrix}$ with $a,b,c,d\in\mathbb{Z}$, $ad-bc = 1$, $a$ and $d$ are equal to $1$ module $p^k$ and $b$ and $c$ are multiples of $p^k$.
How can I show that $|\Gamma _2(p)/\Gamma _2(p^k)|\le p^{4k}$?

By exhibiting a set of at most $N=p^{4k}$ matrices $A_1,\ldots,A_N\in\Gamma_2(p)$ such that for each $B\in\Gamma_2(p)$ we have $A_iB\in\Gamma_2(p^k)$ for some $i$.
Matrices of the form $A_i=I+pM_i$ suggest themselves.
Or much simpler: by counting how many matrices in $\Gamma_2(p)$ and $\Gamma_2(p^k)$ map to the same element of $SL_2(\mathbb Z/p^k\mathbb Z)$ under the canonical projection.
