Divergence of the Harish-Chandra-Howe Integral Formula for Non-connected Groups

Let $G^+$ be (the group of rational points of) a nonconnected, reductive $p$-adic group with connected component $G$, and let $\pi^+$ be an irreducible extension to $G^+$ of a supercuspidal representation $\pi$ of $G$. I will outline a specific example of the divergence of the Harish-Chandra-Howe integral formula for the character of $\pi^+$ on sufficiently regular elements of $G^+$ which do not lie in $G$, and will discuss a work-around in the case of $G = \mathrm{GL}_n$.