  Behind this sophisticated title hides an elementary exercise on Clifford
theory for index two subgroups and self-dual/conjugate-dual representations.
When applied to semi-simple representations of the Weil-Deligne group $W'_F$ of
a non Archimedean local field $F$, and further translated in terms of
representations of $\mathrm{GL_n}(F)$ via the local Langlands correspondence
when $F$ has characteristic zero, it yields various statements concerning the
behaviour of different types of distinction under quadratic base change and
automorphic induction. When $F$ has residual characteristic different from $2$,
combining of one of the simple results that we obtain with the tiviality of
conjugate-orthogonal root numbers (proved by Gan, Gross and Prasad), we recover
without using the LLC a result of Serre on the parity of the Artin conductor of
orthogonal representations of $W'_F$. On the other hand we discuss its parity
for symplectic representations using the LLC and the Prasad and Takloo-Bighash
conjecture.
