Is it safe to run extension cords through walls?

No, it’s not safe, and it’s almost certainly against electrical code. Extension cords are designed for temporary, visible use; running one inside a wall traps heat, hides damage, and creates a serious fire hazard. The proper alternative is to install a new outlet with NM-B cable or use a surface-mounted raceway rated for in-wall use. Don’t risk burning down your house to save money.