Here is a question I came across recently:
If a morphsim in Grp (category of groups) is an epimorphism, then it is a surjective group homomorphsim.
I believe it boils down to show that for any group $G$ and its proper subgroup $H$, there exists a group $G'$ and two group homomorphisms $\phi, \psi: G \to G'$ such that $\phi|_{H} = \psi|_{H}$ but $\phi \neq \psi$. However, I am stuck at this point.
There are also two mirror questions related to this one:
If a morphsim in R-Mod (here $R$ is assumed to be unital) is an epimorphism, then it is a surjective module homomorphsim;
If a morphsim in Ring (category of rings) is an epimorphism, then it is NOT necessarily a surjective ring homomorphsim.
The first I don't know how to prove either while for the second question I found the example of the embedding $\mathbb{Z} \hookrightarrow \mathbb{Q}$. It seems weird to me that between rings there is a distinction between epimophisms and surjective maps while in groups or modules there isn't. Are there some deep reasons behind this?