Let $V$ be a finite-dimensional irreducible representation (complex or $\ell$-adic) of a group $G$ (compact Lie group or algebraic group etc.). Does there always exist a linear character $\rho$ of $G$, such that $V\otimes\rho$ is a self-dual irrep. of $G?$ Namely $V\otimes\rho\simeq(V\otimes\rho)^*.$ If not, is there any necessary/sufficient conditions on $V$ for it to be "twisted self-dual"?
If this is always the case, then in particular, if $G$ has no non-trivial linear characters (e.g. $G$ is a simply-connected compact Lie group or a perfect finite group), then every irrep. of $G$ is self-dual.