  The basic automorphism group ${A}_B(M,F)$ of a Cartan foliation $(M, F)$ is
the quotient group of the automorphism group of $(M, F)$ by the normal
subgroup, which preserves every leaf invariant. For Cartan foliations covered
by fibrations, we find sufficient conditions for the existence of a structure
of a finite-dimensional Lie group in their basic automorphism groups. Estimates
of the dimension of these groups are obtained. For some class of Cartan
foliations with integrable an Ehresmann connection, a method for finding of
basic automorphism groups is specified.
