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I think that there might be some textbooks which introduce the notions of pushout and pullback of a short exact sequence of groups. However, I cannot find any of them.

To be precise, for a given short exact sequence of groups $1\to G'\to G\to G''\to 1$, what are 'the pushout of the sequence by $G'\to H'$' and 'the pullback of the sequence by $G''\to H''$'?

Anyone who can give me the concrete description of the pushout/pullback sequences, it would be very helpful for me.

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It is complicated (in general - perhaps not for, say, finite groups). I found the paper Fibre products, non-positive curvature and decision problems by Baumslag, Bridson, Miller and Short (fibre product=pullback) to be very helpful. If I get time someday I will post an answer, but this is only because I want to learn more and doing so would help. I know very little. Probably someone will post one first I will read it with interest... –  user1729 Jun 17 at 16:23
    
(The fibre product of a short exact sequence $1\rightarrow N\rightarrow\Gamma\rightarrow Q\rightarrow_p 1$ is the group $P=\{(\gamma_1, \gamma_2):p(\gamma_1)=p(\gamma_2)\}\leq\Gamma\times\Gamma$) –  user1729 Jun 17 at 16:25

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