Do page tables used to store kernel stack pointer when context switch happen in kernel mode of the process?

I have two questions; 1. Suppose a user space application/process is running in kernel mode. I understand if a context switch is happened now, the kernel stack pointer of that process is stored in the task_struct. Is that correct? For that, do a PTE (page table entry) created in the page table to map the Kernel Stack Pointer (which is a Virtual address) to the Physical Address? 2. In the case of a kthread, does it have a page table to support context switch?