In this article https://arxiv.org/abs/0905.0071, Jan Essert defines Relative Group Homology $H_*(G,G';M)$ of the groups $G'\leq G$ with coefficients in a $G$-module $M$ as the homology of quotient $$ \frac{F_{\bullet}(G)\otimes_G M}{F_{\bullet}(G')\otimes_{G'} M} $$ where $F_{\bullet}(H)$ is the standard resolution of a group H.
But the article didn't say something about the long exact sequence (but mention it in the proof of a version of Lyndon-Hochschild-Serre spectral sequence)
There is another source for relative group homology?.
EDIT: I forget explicit the long exact sequence, I mean $$ \cdots\to H_n(G',M)\to H_n(G,M)\to H_n(G,G';M)\to H_{n-1}(G',M)\to\cdots $$ EDIT 2: I don't need the short exact seqeuence...
EDIT 3: I found the answer in this book: Kevin P. Knudson - Homology of Linear Groups, pg. 153. Thank you...