Mean Value of Path on the 3-Sphere defined by the Combination of Two Rotations

Question

Suppose that we have a unit quaternion on $q_a\in S^3\subset\mathbb{H}$ and a quaternion that defines a circle on the 3-sphere $q_z(\zeta)\in S^3\subset\mathbb{H}$, where $\zeta\in[0,2\pi]$. Suppose, for the sake of a pedagogical example, that $q_z(\zeta)=(\cos\zeta,\hat{\mathbf{z}}\sin\zeta)$. How do we find the mean of the path on the 3-sphere defined by the variation of $\zeta$, corresponding to the rotation of the reference frame? Taking the normal mean value $\frac{1}{2\pi}\int_0^{2\pi}q_aq_z(\zeta)d\zeta$ gives us an answer off of the path. This should be related in some intuitive way to the SLERP operation, but SLERP defines a different path between $q_a$ and $q_aq_z(\zeta)$: $\int_0^{1}\frac{\sin[(1-t)\arccos(q_a\cdot q_z(\pi))]}{\sin[\arccos(q_a\cdot q_z(\pi))]}q_a+\frac{\sin[t\arccos(q_a\cdot q_z(\pi))]}{\sin[\arccos(q_a\cdot q_z(\pi))]}q_aq_z(\pi)dt$, the average along half the path defined using SLERP, differs at its halfway point: $\frac{1}{\sqrt{2}}(q_a + q_aq_z(\pi))\neq q_aq_z(\frac{\pi}{2})$. Any hints as to take the spherical mean on an arbitrary path of the sphere (which is not necessarily a circle)?