In an earlier paper [Tsukuba J. Math. 17, 461-479 (1993; Zbl 0819.16036)], the authors studied \(G\)-graded coalgebras \(C\) for a group \(G\), and the categories \(\text{gr}^C\) of graded right \(C\)-comodules, \({\mathbf M}^C\) of right \(C\)-comodules, and \({\mathbf M}^{C_1}\) of right \(C_1\)-comodules. In this paper they consider the forgetful functor \(U\) from \(\text{gr}^C\) to \({\mathbf M}^C\) and look at the question of how the properties of \(M \in \text{gr}^C\) are transformed by this functor. They show, for example, that injectivity is preserved. They also study the structure of \(U(\Sigma)\) for a simple object \(\Sigma\) of \(\text{gr}^C\), which is the coalgebra analogue of Clifford theory for graded rings. In this direction they obtain the coalgebra equivalents of the structure results for graded simple \(R\)-modules given for a \(G\)- graded ring \(R\) by the reviewer and C. Năstăsescu [J. Algebra 141, No. 2, 484-504 (1991; Zbl 0742.16022)].