A matrix Paley-Wiener theorem for non-connected $p$-adic reductive groups

Let $F$ be a local non archimedian field of characteristic $0$, and $G$ a non-connected reductive group over $F$. We denote $G^0$ the connected component of the identity and assume the quotient $G/G^0$ is abelian. For $f$ a locally constant compactly supported function on $G$ and $\pi$ a complex smooth representation of $G$, we define the Fourier transform of $f$ evaluated at $\pi$ to be $\pi(f) = \int_{G} f(g) \pi(g) \, dg$, which is an endomorphism of the underlying vector space of $\pi$. We give a description of the image of this Fourier transform map : given, for every $\pi$ in a certain family of induced representations of $G$, an endomorphism $\varphi(\pi)$ of the underlying vector space, we provide necessary and sufficient conditions under which there exists a function $f$ (necessarily unique) such that $\pi(f) = \varphi(\pi)$ for all $\pi$ in the family.