If $G$ and $H$ are two groups, and $\triangleright$ and $\triangleleft$ are a left action and a right action of $H$ on $G$ by group automorphisms such that $$h\triangleright(g\triangleleft h')=(h\triangleright g)\triangleleft h',$$ then we can construct a group $G\bowtie H$ with underlying set $G\times H$ and multiplication such that $$(g,h)\cdot(g',h')=((g\triangleleft h')(h\triangleright g'),hh').$$ This group can also be described as a semidirect product but …
does this particular construction have a name and/or appears in the literature?