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

Stable conjugacy and epipelagic L-packets for Brylinski-Deligne covers of Sp(2n)