  The $K$-groups of the crossed product of the rotation C*-algebra $A_\theta$
by free and amalgamated products of the cyclic groups $\mathbb Z_n$, for
$n=2,3,4,6$, are calculated. The actions here arise from the canonical actions
of these groups on the rotation algebra under the flip, cubic, Fourier, and
hexic automorphisms, respectively. An interesting feature in this study is that
although the inclusion $A_\theta \to A_\theta \rtimes \mathbb Z_n$ induces
injective maps on their $K_0$-groups, the same is not the case for the
inclusions $A_\theta \rtimes \mathbb Z_d \to A_\theta \rtimes \mathbb Z_n$ for
$2\le d < n \le 6$ and $d|n$, which we endeavor to calculate. Further, while
for free products $K_1(A_\theta \rtimes [\mathbb Z_{m} \ast \mathbb Z_{n}]) =
0$, for amalgamated products $K_1(A_\theta \rtimes [\mathbb Z_{m} \ast_{\mathbb
Z_{d}} \mathbb Z_{n}]) = \mathbb Z^k$ is non-vanishing ($k=1,2$).
