Could a Pet 'Fancy Mouse' live outside in a Rabbit Hutch?

I've recently been interested in buying a fancy mouse. I have an indoor cage for one all ready for one to settle in, however for the summer, I was thinking I could build a smaller version of a rabbit hutch with much smaller wire mesh on the door (so it can't escape) and hope that it can live outdoors for the summer months. I appreciate all the answers, so thank-you for your time.