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In Humphrey's Group theory, Chapter 21 gives the construction of Central Extensions. [A group G is a central extension of N by H with Z(G)=N.] You could use that construction to build appropriate examples.

Wreath products are also semidirect products. In Weinstein's Examples of Groups, Sec 4.4 on wreath products, Theorem 4.4.7 constructs and proves the restricted wreath product W=(G rwr A) such that Z(W) = Z (diag($G^A$) x {1}) where diag is the diagonal subgroup (result applies only if A is finite)

Meldrum's book on Wreath Products Chap 1 might provide more constructions.

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In Humphrey's Group theory, Chapter 21 gives the construction of Central Extensions. [A group G is a central extension of N by H with Z(G)=N.] You could use that construction to build appropriate examples.

In Weinstein's Examples of Groups, Sec 4.4 on wreath products, Theorem 4.4.7 constructs and proves the restricted wreath product W=(G rwr A) such that Z(W) = Z (diag($G^A$) x {1}) where diag is the diagonal subgroup (result applies only if A is finite)

Meldrum's book on Wreath Products Chap 1 might provide more constructions.