We prove upper semicontinuity of the isotropy subgroups and identity components of automorphism groups of taut manifolds with respect to the topology induced by a distance function on the sets of pointed taut manifolds which is defined in terms of certain extremal problems of holomorphic mappings. Namely, it is proved that given a pointed taut manifold \(M\), for any pointed taut manifold \(M'\) sufficiently close to \(M\) in the topology mentioned above, the isotropy subgroup at the distinguished point (resp. identity component) of the automorphism group of \(M'\) is a subgroup of the corresponding group of \(D\). For the part of the above assertion about the identity component of the automorphism group, we need to assume that the automorphism group of \(M\) is compact.