Help...what to do about cat

[Ladygaggia]

I've come home from holiday and was met by a black cat that ran in off the street and seemed friendly. It snuck in the house and my son gave it some ham.
I gave it some water and put it outside. It is really very friendly, slightly skinny, but I'm no cat expert.

I gave it some mackerel later as it was still sneaking in.

I have put a box outside with a blanket in, and I've just peaked and it's snuggled up in there.

I can't have it in the house as I rent... I've posted on Facebook about a found cat...what else should I do?

Put it outside?
Stop feeding it?
Or take it to vet to check for chip?
What makes you think it's a stray?

It is outside, in the box. Wouldn't it go home if it had one?
I've never owned a pet bigger than a hamster...I haven't a clue