  We prove several dimension formulas for spaces of scalar-valued Siegel
modular forms of degree $2$ with respect to certain congruence subgroups of
level $4$. In case of cusp forms, all modular forms considered originate from
cuspidal automorphic representations of $\mathrm{GSp}(4,\mathbb{A})$ whose
local component at $p=2$ admits non-zero fixed vectors under the principal
congruence subgroup of level $2$. Using known dimension formulas combined with
dimensions of spaces of fixed vectors in local representations at $p=2$, we
obtain formulas for the number of relevant automorphic representations. These
in turn lead to new dimension formulas, in particular for Siegel modular forms
with respect to the Klingen congruence subgroup of level $4$.
