  Let $F$ be a local field of characteristic not $2$. We propose a definition
of stable conjugacy for all the covering groups of $\mathrm{Sp}(2n,F)$
constructed by Brylinski and Deligne, whose degree we denote by $m$. To support
this notion, we follow Kaletha's approach to construct genuine epipelagic
$L$-packets for such covers in the non-archimedean case with $p \nmid 2m$, or
some weaker variant when $4 \mid m$; we also prove the stability of packets
when $F \supset \mathbb{Q}_p$ with $p$ large. When $m=2$, the stable conjugacy
reduces to that defined by J. Adams, and the epipelagic $L$-packets coincide
with those obtained by $\Theta$-correspondence. This fits within Weissman's
formalism of L-groups. For $n=1$ and $m$ even, it is also compatible with the
transfer factors proposed by K. Hiraga and T. Ikeda.
