I am going to describe the diagram since I do not know how to do them on latex. Say $H$ is a subgroup of $G$ and we have $H\rightarrow G$ via inclusion (call this map $i$), and $H\rightarrow 1$ via the trivial map (call this map $z$). We want to find the pushout of it.
I think that the pushout will be $1$ because the map $f_1:G\rightarrow 1$ and the map $f_2:1\rightarrow 1$ are such that $f_2(z(h))=f_1(i(h))$ for all $h\in H$. Further, since $1$ is an initial object in the category of groups then we would be done.
Is the above correct?