There are times when a councillor is appointed as parish council representative on outside bodies - for example public transport user forums or governor for a school so they can act on their own. However, they cannot make decisions on behalf of the council, so there is a limit to their delegation - they have the authority to represent but not act on behalf of the council in these instances.
I have some Councillors who do research to present back to the council on matters that they have knowledge and expertise on at the start of projects but once the project is formally adopted by the council, then it is passed to me to action and take forward.