do satellites use summertime... tl;dr: no.

they use Temps Atomique International (TAI) [1,2].

#GNSS time vs local time

#GPS time is a continuous time scale with no leap seconds, and #GALILEO System Time is also continuous and not affected by daylight saving time.
#BeiDou the same.

#GLONASS is likewise a GNSS time reference tied to the system’s own timing, not to seasonal civil clock changes BUT the use *leap seconds*, that hurts.

[1] https://www.tualcom.com/gnss-system-times/
[2] https://www.iers.org/SharedDocs/Glossareintraege/EN/I/int_at_time_tai