Suppose \(K\) is a compact Lie group and \(X\) is a Stein manifold on which \(K\) and its complexification \(K^{\mathbb{C}_ C}\) acts as groups of biholomorphic transformations. Assume \(\Omega\) is a \(K\)-invariant domain in \(X\). The present paper is concerned with the problem of determining the envelope of holomorphy \(E(\Omega)\) of \(\Omega\). Using results of \textit{D. Snow} [Math. Ann. 259, 79-97 (1982; Zbl 0509.32021)] and \textit{P. Heinzner} [Math. Ann. 289, No. 4, 631-662 (1991; Zbl 0728.32010)] the authors construct a Stein manifold \(U^*\) together with a holomorphic \(K^{\mathbb{C} c}\)-action such that \(E(\Omega)\) is schlicht and orbit convex in \(U^*\). Under some assumptions of a technical nature the approach of \textit{O. S. Rothaus} [Princeton Math. Ser. 31, P.U. Press, Princeton, 309- 317 (1970; Zbl 0212.108)], generalized to the present setting, is used to give an explicit description of the envelope of holomorphy as the interior of the zero set of a certain largest plurisubharmonic minorant.