If you describe a task or job as invidious, you mean that it is unpleasant because it is likely to make you unpopular. The local authority could find itself in the invidious position of having to refuse
An invidious comparison or choice between two things is an unfair one because the two things are very different or are equally good or bad. Police officers fear invidious comparisons. unpleasant, especially because it is likely to offend people or make you unpopular (invidiosus, from invidia )