I've just discovered this spider and its eggs inside a window frame in my house. I would be grateful if someone could help me to identify whether it is a false widow spider and provide any guidance on extermination.
It does look like Steatoda nobilis (noble false widow).
Open the window & shoo them out, start with mum (these spiders are usually very docile, but I wouldn't fancy disturbing a mother & her newly hatched young). The (harmless) spiderlings will disperse naturally, they don't really want to be couped up indoors & tripping over each other as they get bigger.
Alternatively, let them disperse of their own acccord? Remove any spiderlings you see roaming around indoors. The female, if left to her own devices, will probably stick to her web.
There isn't really any need to exterminate them, but obviously, that is entirely your call.