  Let $\mathfrak{o}$ be a compact discrete valuation ring with maximal ideal
$\mathfrak{p}$ such that the finite residue field $\mathfrak{o}/\mathfrak{p}$
has characteristic $p.$ For $r\geq2$ and $p=2,$ we obtain the branching rules
for the restriction of a regular representation of
$\mathrm{GL}_2(\mathfrak{o}/\mathfrak{p}^r)$ to
$\mathrm{SL}_2(\mathfrak{o}/\mathfrak{p}^r).$ These results have different
behaviour than that of the known case of $p\neq2.$
