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@Charles 9

In such a situation, phones would likely be internally routed through a PBX or the like. In which case, the call would be routed through the internal network before being forwarded outside

Yeah, that's one way to do it. Company in one location contracts out calling customers (I'm talking about calls the customers actually want, not unwanted calls) to a call centre in another location (or possibly more than one call centre). So when the call centre makes a call, it dials a special number for the company that contracted it, a number that is a hunt group for several (perhaps large values of "several") lines at the company itself. Those lines go to a PBX, which then forwards the call to the customer over yet more lines, all to make it look like the call came from the company itself. That's some rather expensive tromboning, requiring twice the number of lines as actual calls and a PBX.

Or, the company authorizes the call centre to make calls on its behalf using the appropriate calling ID for the company. Legal, legit and moral (again, assuming these are calls the customers want to receive).

Either way the customer gets a call that identifies itself as being from the company (but is actually from a call centre). One way is a lot more expensive than the other.

Note: "Lost all faith..." didn't explicitly say that this was what the call centre was doing but he/she/it left enough clues that it was easy to figure out. Yet few people managed to do so. Not even after I responded to him and left a clue that people were missing something.