Cloudy is an Enigma

We have had many new homeless cats show up over recent months. This one is named Cloudy. We can handle him but when we invite him into our house he looks around for awhile then leaves. I feel certain he is homeless based on the volume of food he eats. We are reluctant to just put him into a carrier and get him adopted since maybe he's more comfy outdoors (tho that is beyond my understanding). The last two cats that were approachable which we tried to bring inside became fiercely resistant.  Yikes!