Suppose we have a function define on unit sphere which is parametrized by $u(\theta, \phi)$. Now I apply an automorphism $A: S^2 \rightarrow S^2$ on the sphere that rotates the image of the function $u$. This would gives a new function $u_A$ which theoretically could be written as $u_A(\theta, \phi) = Au(\theta, \phi)$.
My question is: do we have a general formula for $u_A$ on spherical coordinate? For example, if the automorphism is rotation around $z$ axis by angle $\alpha$, then the new function would be $u_A=u(\theta, \phi-\alpha)$. However, if the automorphism is rotation around $y$ axis, then I cannot think of simple form for $u_A$.
I understand that it is possible to parametrize everything in Euclidean coordinate -- but the formula seems to be quite cumbersome. My motivation for this question is: I am studying diffusion on sphere and I tried to perform some superposition of fundamental solutions of multiple initial diffusion sources located on different $(\theta, \phi)$. Thanks in advance!