Cancellation theorem in group theory (for direct product) says that if $B$ is a finite group and $A \times B \simeq A_1 \times B_1$ and $B \simeq B_1$ then $A \simeq A_1.$
Of course, if $B$ is not finite, the result is absurd, even for finitely presented groups (Here is an example by Steve)
I wonder whether the cancellation
theorem holds for different products
(in finite or infinite cases), such as
semi-direct product, free product,
fiber product over a given group,
Zappa-Szep product (knit product),
Wreath product.
Best Answer
For a straightforward example that the subgroup doing the acting in a semidirect product cannot automatically be cancelled (and as semidirect products are knit products, also addresses that case), the dihedral group of order $8$ can be written as a semidirect product in two different ways: $C_4\rtimes C_2$ (with the action being inversion) and $(C_2\times C_2)\rtimes C_2$ (with the action being swap coordinates).