  We study the graded polynomial identities with a homogeneous involution on
the algebra of upper triangular matrices endowed with a fine group grading. We
compute their polynomial identities and a basis of the relatively free algebra,
considering an arbitrary base field. We obtain the asymptotic behaviour of the
codimension sequence when the characteristic of the base field is zero. As a
consequence, we compute the exponent and the second exponent of the same
algebra endowed with any group grading and any homogeneous involution.
