Homology and semidirect products If $G=N\rtimes H$ what is the relation between the second integral homology groups (Schur multipliers) of $G,N$ and $H$.  

The natural map $H_2(G) \rightarrow H_2(H)$ is surjective for a split extension. Apart from that, you cannot say anything very definitive. There is a natural map $H_2(N) \rightarrow H_2(G)$, but that is not usually injective. Also, the image of that map is not in general equal to the kernel of $H_2(G) \rightarrow H_2(H)$. There is another section of $H_2(G)$ coming (roughly) from commutators in $[N,H]$.
The Lyndon-Hochschild-Serre Spectral Sequence provides a theoretical background for all of this, but it does not necessarily help with calculations in specific examples.
Of course, for the direct product, $G = N \times H$, we have $H_2(G) \cong H_2(N) \oplus H_2(H) \oplus (N \otimes H)$.
