  We characterize when a C*-cover admits a C*-dynamical extension of dynamics
on an operator algebra in terms of the boundary ideal structure for the
operator algebra in its maximal representation and show that the C*-covers that
admit such an extension form a complete lattice. We study dynamical systems
arising from groups acting via inner automorphisms in a C*-cover and produce an
example of a C*-cover that admits no extension of dynamics on a
finite-dimensional non-self-adjoint operator algebra. We construct a partial
action on a class of C*-covers that recovers the crossed product of an operator
algebra as a subalgebra of the partial crossed product, even when the C*-cover
admits no dynamical extension.
