I you really meant H(s) transfer function or H(jw) and not H(t)
I think that Laplace transform should be applied only to LTI systems. So you will not get a valid h(s) if you try to get it from a non LTI. I someone gives you H(s) then implies that it was a result of a LTI system.
Same with Fourier transform that will give you H(jw).