  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.
