Let
S
S
and
T
T
be semigroups,
S\text {\textcircled {𝜏
}} T a semidirect product, and
F
F
a
C
∗
{C^ \ast }
-algebra of bounded, complex-valued functions on
S\text {\textcircled {𝜏
}} T. Necessary and sufficient conditions are given for the
F
F
-compactification of
S\text {\textcircled {𝜏
}} T to be expressible as a semidirect product of compactifications of
S
S
and
T
T
. This result is used to show that the strongly almost periodic compactification of
S\text {\textcircled {𝜏
}} T is a semidirect product and that, in certain general cases, the analogous statement holds for the almost periodic compactification and the left uniformly continuous compactification of
S\text {\textcircled {𝜏
}} T. Applications are made to wreath products.