When you call a system call such as fork in process X, the kernel is said to be executing in process context. So, fork can be said to be running in pro
The scheduler take care of all processes, so does not run inside one process.
Of course, when e.g. a process is scheduled out because of a clock interrupt, some process was running (and later, another one is scheduled).
You cannot view all the kernel as running for processes (only system calls are).