The motivation for this question is that I am trying to Read Basic Algebra 2 (despite a somewhat weak background for the book) and if this is true then it is fairly easy to show that monics in Grp are injective and epics are surjective.
In linear algebra/functional analysis you can project an element onto the subspace orthogonal to a given subspace. I wonder if something like this holds in group theory.
Another Question: Does this hold for modules? Given a $R$-module $M$ and a $R$-submodule $N$ is there a homomorphism $f: M \rightarrow M$ s.t $f(N) = \operatorname{id}$?
edit: also wanted $f(x) \neq \operatorname{id}$ if $x$ is not in $H$.