For two short exact sequences of say, finitely generated modules of some ring $$0\rightarrow N\xrightarrow{a} R\xrightarrow{b} M\rightarrow0, \qquad 0\rightarrow K\xrightarrow{a'}R\xrightarrow{b'}L\rightarrow0.$$
Let $\phi\in \operatorname{Aut}(R),$ and $\phi$ acts on those two short exact sequences by $\phi\cdot a=\phi\circ a, \phi\cdot a'=\phi\circ a',\phi\cdot b=b\circ\phi^{-1},\phi\cdot b'=b'\circ\phi^{-1}.$
My question is that the book says the cardinality of stabilizer of this action is the same as the cardinality of $\operatorname{Hom}(\operatorname{Coker} b'a, \operatorname{Ker} b'a).$ But I don't know why, can somebody give me some idea?