rt.representation theory - extending cusp forms

I believe the answer should be yes, by some version of the following sketch of an argument:

(Note: by restriction of scalars, I regard all groups as being defined over $mathbb Q$,
and I write ${mathbb A}$ for the adeles of $mathbb Q$.)

We are given $V_{pi} subset Cusp(G(F)backslash G({mathbb A}_F)).$

Let $tilde{C}$ denote the maximal $mathbb Q$-split torus in the centre of $tilde{G}$
(this is just a copy of $mathbb G_m$), and write $C = tilde{C}cap G$. (I guess
this is just $pm 1$?)

Now $C(mathbb A)$ acts on $V_{pi}$ through some character $chi$ of $(mathbb A)/C(mathbb Q)$. Choose an extension
$tilde{chi}$ of $chi$ to a character of $tilde{C}(mathbb A)/tilde{C}(mathbb Q)$,
and regard $V_{pi}$ as a representation of $tilde{C} G$ by have $tilde{C}$ act
through $tilde{chi}$. Since $tilde{C} G$ is normal and Zariksi open in $tilde{G}$,
we should be able to further extend the $tilde{C} G(mathbb A)$-action on $V_{pi}$
to an action of $tilde{G}(mathbb Q)tilde{C}G(mathbb A).$

Now if we consider $Ind_{tilde{G}(mathbb Q)tilde{C} G(mathbb A)}^{tilde{G}(mathbb A)} V_{pi},$ we should be able to
find a cupsidal representation $V_{tilde{pi}}$ of the form you want (with $tilde{C}(A)$ acting via $tilde{chi}$).

The intuition is that automorphic forms on $G(mathbb A)$ are $Ind_{G(mathbb Q)}^{G(mathbb A)} 1,$
and similarly for $tilde{G}$. We will consider variants of this formula that takes into account central characters, and think about how to compare them for $G$ and $tilde{G}$.

Inside the automorphic forms, we have those where $C(mathbb A)$ acts by $chi$; this we can
write as $Ind_{G(mathbb Q)C(mathbb A)}^{G(mathbb A)} chi$, and then rewrite as
$Ind_{tilde{G}(mathbb Q)tilde{C}(mathbb A)}^{tilde{G}(mathbb Q)tilde{C}G(mathbb A)} tilde{chi}.$ This is where $V_{pi}$ lives, once we extend it to a repreresentation
of $tilde{G}(mathbb Q)tilde{C}G(mathbb A)$ as above.

Now the automorphic forms on $tilde{G}(mathbb A)$ with central character $tilde{chi}$
are $Ind_{tilde{G}(mathbb Q)tilde{C}(mathbb A)}^{tilde{G}(mathbb A)} tilde{chi},$
which we can rewrite as
$Ind_{tilde{G}(mathbb Q) tilde{C}G(mathbb A)}^{tilde{G}(mathbb A)}
Ind_{tilde{G}(mathbb Q)tilde{C}(mathbb A)}^{tilde{G}(mathbb Q)tilde{C}G(mathbb A)}
tilde{chi}.$ This thus contains $Ind_{tilde{G}(mathbb Q)tilde{C}G(mathbb A)}^{tilde{G}(mathbb A)}V_{pi}$ inside it, and so an irreducible constituent of the latter
should be a $V_{tilde{pi}}$ whose restriction (as a space of functions) to $G(mathbb A)$ contains $V_{pi}$.

What I have just discussed is the analogue for $G$ and $tilde{G}$ of the relation between automorphic forms on $SL_2$ and $GL_2$ discussed e.g. in Langlands--Labesse. Hopefully I haven't blundered!