How does permafrost not thaw in the summer?

Yes, the first few inches thaw. This allows short plants without large roots to thrive during the short summer. The soil freezes from the top down. The limit of permafrost is called the frost front, and advances downward. It can be several feet thick to hundreds of feet thick in some parts of Alaska. During the summer thaw, the soil melts from the top down. Because the summer is so short, this melting doesn't get very far, and just refreezes in the winter. All the while, even during summer, the frost front advances below ground, and the permafrost becomes deeper. The soil is an insulator, and the temperature takes awhile to freeze or warm up, so the frost front and the summer thaw both advance very slowly. The water in the uppermost layers can freeze rather quickly, though, because it is exposed to the air.