Extension of a faithful representation from an open normal subgroup

Let $H$ be an open normal subgroup of a compact topological group $G$, and let $\tau :H\to {\mathrm{GL}}_m({\mathbb{Z}}_p)$ be a continuous faithful representation. Choose representatives $t_j$ for the finite left cosets $G/H$. Write $g{t}_j=t_i{h}_{ij}$ and act on the $j$-th summand of ${⨁}_j{\mathbb{Z}}_p^m$ by $\tau (h_{ij})$ into the $i$-th summand. The cocycle identity proves this is a continuous group representation. An element outside $H$ has a nontrivial coset permutation, while an element of $H$ acting trivially has $\tau (g)=1$ on the identity-coset summand. Hence the representation of $G$ is faithful and integral.