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Given any group $G$, the multiple holomorph $\mathrm{NHol}(G)$ is thenormalizer of the holomorph $\mathrm{Hol}(G) = \rho(G)\rtimes \mathrm{Aut}(G)$in the group of all permutations of $G$, where $\rho$ denotes the right regularrepresentation. The quotient $T(G) = \mathrm{NHol}(G)/\mathrm{Hol}G)$ has ordera power of $2$ in many of the known cases, but there are exceptions. We shallgive a new method of constructing elements (of odd order) in $T(G)$ when$G=A\rtimes C_d$, where $A$ is a group of finite exponent coprime to $d$ and$C_d$ is the cyclic group of order $d$.

The multiple holomorph of a semidirect product of groups having coprime exponents