A group G is an extension of A by B (A,B groups) if there exists a normal subgroup of G (call it N) such that A is isomorphic to N, and G/N is isomorphic to B. Is there a simple way to identify ALL such extensions, if A, B are small groups (order 4 or less would dignify "small"). Simple or not, how is it done? -I've looked throughout the web, and cannot find much on "group extensions". Furthermore, is there a way to say how many extensions of A by B exist? I decided to try a simple example (finding as many non-isomorphic extensions of Z_2 by Z_2 (integers mod 2), and obviously theres the Z_2 X Z_2 extension, and there's Z_4 . What other possibilities are there?