Twisted partial actions and extensions of semilattices of groups by groups

We introduce the concept of an extension of a semilattice of groups $A$ by a group $G$ and describe all the extensions of this type which are equivalent to the crossed products $A{\ast }_{\mathrm{\Theta }}G$ by twisted partial actions $\mathrm{\Theta }$ of $G$ on $A$. As a consequence, we establish a one-to-one correspondence, up to an isomorphism, between twisted partial actions of groups on semilattices of groups and so-called Sieben twisted modules over $E$-unitary inverse semigroups.