  $HC_*(A \rtimes G)$ is the cyclic homology of the crossed product algebra $A
\rtimes G.$ For any $g \epsilon G$ we will define a homomorphism from
$HC_*^g(A),$ the twisted cylic homology of $A$ with respect to $g,$ to $HC_*(A
\rtimes G).$ If $G$ is the finite cyclic group generated by $g$ and $|G|=r$ is
invertible in $k,$ then $HC_*(A \rtimes G)$ will be isomorphic to a direct sum
of $r$ copies of $HC_*^g(A).$ For the case where $|G|$ is finite and $Q \subset
k$ we will generalize the Karoubi and Connes periodicity exact sequences for
$HC_*^g(A)$ to Karoubi and Connes periodicity exact sequences for $HC_*(A
\rtimes G)$ .
