Previous work of the authors [Funct. Anal. Appl, 27, No. 1, 25--32 (1993); translation from Funkts. Anal. Prilozh. 27, No. 1, 25--32 (1993; Zbl 0804.22011); Trans. Am. Math. Soc. 351, 477--495 (1999; Zbl 0911.22005); Linear Multilinear Algebra 36, No. 2, 79--101 (1993; Zbl 0797.15010)], in the setting of compact groups, introduced the wrapping map \(\Phi\). This map associates, to each Ad-invariant distribution \(\mu\) of compact support on the Lie algebra \(\mathfrak g\), a central distribution \(\Phi_\mu\) on the Lie group \(G\), via the formula, for \(f\in C_c^\infty(G)\), \[ \langle \Phi_\mu, f\rangle = \langle\mu, j\cdot f\circ\exp\rangle, \] where \(j\) is the square root of the Jacobian of \(\exp: {\mathfrak g}\mapsto G\). The remarkable thing about \(\Phi\) is that it provides a convolution homomorphism between the Euclidean convolution structure on \(\mathfrak g\) and the group convolution on \(G\). In the paper under review the authors extend their results to compact times vector semidirect products. In particular, they define the convolution of noncompact coadjoint orbits and recover the character formulae and Plancherel formula of Lipsman.