Uniqueness up to Inner Automorphism of Regular Exact Borel Subalgebras

K\"ulshammer, K\"onig and Ovsienko proved that for any quasi-hereditary algebra (A,≤A) there exists a Morita equivalent quasi-hereditary algebra (R, ≤R) containing a basic exact Borel subalgebra B. The obtained Borel subalgebra is in fact a regular exact Borel subalgebra. Later, Conde showed that given a quasi-hereditary algebra (R,≤R) with a basic regular exact Borel subalgebra B and a Morita equivalent quasi-hereditary algebra (R',≤R') with a basic regular exact Borel subalgebra $B'$, the algebras R and $R'$ are isomorphic, and K\"ulshammer and Miemietz showed that there is even an isomorphism φ:R→ R' such that φ(B)=B'. In this article, we show that if $R=R'$, then φ can be chosen to be an inner automorphism. Moreover, instead of just proving this for regular exact Borel subalgebras of quasi-hereditary algebras, we generalize this to an appropriate class of subalgebras of arbitrary finite-dimensional algebras. As an application, we show that if (A, ≤A) is a finite-dimensional algebra and G is a finite group acting on A via automorphisms, then under some natural compatibility conditions, there is a Morita equivalent quasi-hereditary algebra (R, ≤R) with a basic regular exact Borel subalgebra B such that $g(B)=B$ for every g∈ G.