Lifting Automorphisms of Quotients by Central Subgroups

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Given a finitely presented group G, we wish to explore the conditions under which automorphisms of quotients G/N can be lifted to automorphisms of G. We discover that in the case where N is a central subgroup of G, the question of lifting can be reduced to solving a certain matrix equation. We then use the techniques developed to show that Inn(G) is not characteristic in Aut(G), where G is a metacyclic group of order pⁿ, p≠ 2.