It's not hard to imagine early group theorists getting the inspiration for the semidirect product because after you've seen a few examples of finite nonabelian groups, the pattern starts to emerge on its own.
But who first codified the definition, explicitly proposed looking at a mapping $\varphi :H\to Aut(N)$, and showed that $\langle n, h\rangle\langle n', h'\rangle = \langle n\varphi_{h}(n'), hh'\rangle$ gives a group operation on the product set $N\times H$, and when?
I'd be interested in any leads on any part of this: earlier prefigurings and special cases; later distillations; who coined the name; etc.
This question, who invented the semidirect product (and, the holomorph for that matter) , interested me for a long time, too. Two people might be good candidates: G.A. Miller and O. Hoelder. Miller wrote papers on the holomorphs of cyclic groups early in the 1900's and he was a contemporary of Hoelder (a pioneer in considering automorphisms).