  In a physical system undergoing a continuous quantum phase transition,
spontaneous symmetry breaking occurs when certain symmetries of the Hamiltonian
fail to be preserved in the ground state. In the traditional Landau theory, a
symmetry group can break down to any subgroup. However, this no longer holds
across a continuous phase transition driven by anyon condensation in symmetry
enriched topological orders (SETOs). For a SETO described by a $G$-crossed
braided extension $\mathcal{C}\subseteq \mathcal{C}^{\times}_{G}$, we show that
physical considerations require that a connected \'etale algebra $A\in
\mathcal{C}$ admit a $G$-equivariant algebra structure for symmetry to be
preserved under condensation of $A$. Given any categorical action
$\underline{G}\rightarrow \underline{\sf Aut}_{\otimes}^{\sf br}(\mathcal{C})$
such that $g(A)\cong A$ for all $g\in G$, we show there is a short exact
sequence whose splittings correspond to $G$-equivariant algebra structures. The
non-splitting of this sequence forces spontaneous symmetry breaking under
condensation of $A$. Furthermore, we show that if symmetry is preserved, there
is a canonically associated SETO of $\mathcal{C}^{\operatorname{loc}}_{A}$, and
gauging this symmetry commutes with anyon condensation.
