Well, nobody broke a law by entering after permission was given. But, they may have broken the law if they impersonated police officers (and how that law is drafted I do not know). They may also have broken the law (or, alternatively, the officers may face some penalty) for misuse of police resources -- I don't really know what kinds of rules there are about getting police to aid in private investigations, but I imagine there are some.
In any case, the person in question certainly didn't have to let these people in, and probably shouldn't have. But, it can be tough to justify refusing entry to the police if you think you haven't done anything wrong.
If standing next to a plainclothes officer who identifies himself as police is considered impersonating a police officer something is wrong with the law.
It sounds sketchy but I doubt anyone broke the law here.
You probably have never seen this happen, hence your naiveté. When a bunch of cops show up at your door, each of them doesn't identify themselves separately as a cop. The leader will usually announce, "SFPD! We are here to investigate blah blah blah...", and probably show a badge. His other comrades will just stand there. They don't go down the conga line, each swearing an oath that they are cops. It is assumed that the first announcement is enough to establish authority.
Suppose you go on to the gridiron at the start of a match dressed as one of the team - you're just there right, not impersonating a member of the team? Anyone who things you're in the team is just mistaken, the team shouldn't need to make it clear?
In any case, the person in question certainly didn't have to let these people in, and probably shouldn't have. But, it can be tough to justify refusing entry to the police if you think you haven't done anything wrong.