Take, for example, the Fregean explication of the concept of concept.
It follows from the considerations of Dummett (1974), that even if we consider that of the Fregean terms which is really closer to the intuitive concept of meaning, namely his Sinn (sense), we are likely to encounter a parallel ambiguity, for Fregean senses have come to be taken to play two incompatible roles: to explicate what a linguistic agent grasps when she grasps words, and to determine the corresponding Bedeutung, i.e. extension.
Take for example the large body of studies concerning the nature of definite and indefinite descriptions, which have persuasively shown that to see these locutions directly in terms of classical, Fregean quantification is inadequate and may be severely misguiding.
The Fregean split of meaning into Sinn and Bedeutung is well known, but we can go even further.
Carnap (1947) suggested that an intension behaves in many respects like a Fregean sense, the aspect of an expression's meaning that corresponds to its cognitive significance.
For example, it is cognitively significant that all renates are cordates and vice versa (this was a nontrivial empirical discovery about the world), so that 'renate' and 'cordate' should have different Fregean senses.
If this were the case, the distinction between intension and extension could be seen as a sort of vindication of a Fregean distinction between sense and reference.
One might even speculate that an expression's diagonal intension behaves in some respects like a Fregean sense, in a way that might vindicate Carnap's project.
So at least in these domains, character behaves a little like a Fregean sense.
If this is correct, then primary intensions behave in these cases in a manner somewhat reminiscent of a Fregean sense.