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Let $G$ be a finite group, $X$ be a compact $G$-space. In this note we studythe $(\mathbb{Z}_ + \times\mathbb{Z}/2\mathbb{Z})$-graded algebra$$\mathcal{F}^q_G(X) = \bigoplus_{n\geq0} q^n \cdotK_{G\wr\mathfrak{S}_n}(X^n)\otimes\mathbb{C},$$ defined in terms of equivariantK-theory with respect to wreath products as a symmetric algebra. Morespecifically, let $H$ be another finite group and $Y$ be a compact $H$-space,we give a decomposition of $\mathcal{F}^q_{G\times H}(X\times Y)$ in terms of$\mathcal{F}^q_G(X)$ and $\mathcal{F}^q_H(Y)$. For this, we need to study therepresentation theory of pullbacks of groups. We discuss also some applicationsof the above result to equivariant connective K-homology.

Induced character in equivariant K-theory and wreath products