  We prove a version of the weight part of Serre's conjecture for mod $p$
Galois representations attached to automorphic forms on rank 2 unitary groups
which are non-split at $p$. More precisely, let $F/F^+$ denote a CM extension
of a totally real field such that every place of $F^+$ above $p$ is unramified
and inert in $F$, and let $\overline{r}: \textrm{Gal}(\overline{F^+}/F^+)
\longrightarrow {}^C\mathbf{U}_2(\overline{\mathbb{F}}_p)$ be a Galois
parameter valued in the $C$-group of a rank 2 unitary group attached to
$F/F^+$. We assume that $\overline{r}$ is semisimple and sufficiently generic
at all places above $p$. Using base change techniques and (a strengthened
version of) the Taylor-Wiles-Kisin conditions, we prove that the set of Serre
weights in which $\overline{r}$ is modular agrees with the set of Serre weights
predicted by Gee-Herzig-Savitt.
