  We analyze Hecke pairs (G,H) and the associated Hecke algebra when G is a
semidirect product N x Q and H = M x R for subgroups M of N and R of Q with M
normal in N. Conditions are given in terms of N, Q, M, and R which are
equivalent to the Hecke condition on (G,H), and the Schlichting completion of
(G,H) is identified in terms of completions of N, Q, M, and R. Our main result
shows that (assuming (G,H) coincides with its Schlichting completion) when R is
normal in Q, the closure of the Hecke algebra in C*(G) is Morita-Rieffel
equivalent to a crossed product I x Q/R, where I is a certain ideal in the
fixed-point algebra C*(N)^R. Several concrete examples are given illustrating
and applying our techniques, including some involving subgroups of GL(2,K)
acting on K^2, where K = Q or K = Z[1/p].
