Thinking the stray cat might like to come inside where the ac is

There is a stray cat in the neighborhood that I think I am going to adopt. She is a beautiful grey cat with bright green eyes. I have gotten it where she will come to me when I make a clicking noise with my mouth. I give her daily pets, sometimes two times a day. I also set out water and food for her. I have decided she is my cat now that she hangs around the house. I am debating on making her an indoor and outdoor cat. But that is quite a jump. Right now she roams around free and I pet her when I want. If I let her inside, she is fully my cat and can do some damage. I am considering popping her into the house due to the outside heat is in the 100s everyday. I have my air conditioner running non stop. Even with the AC blasting, I am still hot indoors. I can’t imagine the poor cat outside. I notice that she does not move around very much and sleeps in the shade. She is probably trying to conserve energy. The water seems to help, but I can’t help but think she would love the AC. It seems mean watching her walk about outside when I am enjoying the cool air conditioning. How much damage can one little cat do? Maybe she would be more friendly to me if I let her get into the AC too. Right now she is a hit and miss if she wants to scratch me.

a/c install