$\newcommand{\GL}{\mathrm{GL}}\newcommand{\dbZ}{\mathbb{Z}}\newcommand{\dbF}{\mathbb{F}}\newcommand{\dbN}{\mathbb{N}}$ I have a small question regarding closed subgroup of $\GL_n(\dbZ_p)$.
Fix $n\in \dbN$ and let $p$ be a fixed prime. Let $\eta:\GL_n(\dbZ_p)\to\GL_n(\dbF_p)$ denote the coordinate-wise reduction-map (it can be easily verified to be well defined and a homomorphism of these two groups).
Suppose we have two closed (with respect to $p$-adic topology) and infinite subgroups $G, H\in\GL_n(\dbZ_p)$, which have the same image under $\eta$, i.e. $\eta(G)=\eta(H)$.
What can said in this case about $G$ and $H$? I know one can not expect them to be equal (e.g. $G=1+p M_n(\dbZ_p)$ and $H=1+p^k M_n(\dbZ_p)$ have the same image mod $p$).
Can it be proved for example that in such a case $G$ and $H$ are commensurable? perhaps under some additional assumptions regarding these groups? Is this question somewhat easier if it is also known that $G$ and $H$ are the groups of points over $\dbZ_p$ of algebric groups defined over $\dbZ_p$?
I'm kind of in the dark about what exactly I'm looking for with this question, so would appreciate any reference or hint which might be relevant.
Thank you.