On the number of epi-, mono-, and homomorphisms of groups

It is known that the number of homomorphisms from a group $F$ to a group $G$ is divisible by the greatest common divisor of the order of $G$ and the exponent of $F/[F,F]$. We investigate the number of homomorphisms satisfying some natural conditions such as injectivity or surjectivity. The simplest nontrivial corollary of our results is the following fact: {\it in any finite group, the number of generating pairs $(x,y)$ such that $x^3=1=y^5$, is a multiple of the greatest common divisor of 15 and the order of the group $[G,G]\cdot\{g^{15}\;|\;g\in G\}$.Comment: 5 pages. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.htm . V2: minor corrections. arXiv admin note: text overlap with arXiv:1806.0887