  We study $L^p\to L^r$ estimates for restricted averaging operators related to
algebraic varieties $V$ of $d$-dimensional vector spaces over finite fields
$\mathbb F_q$ with $q$ elements. We observe properties of both the Fourier
restriction operator and the averaging operator over $V\subset \mathbb F_q^d.$
As a consequence, we obtain optimal results on the restricted averaging
problems for spheres and paraboloids in dimensions $d\ge2,$ and cones in odd
dimensions $d\ge 3.$ In addition, when the variety $V$ is a cone lying in an
even dimensional vector space over $\mathbb F_q$ and $-1$ is a square number in
$\mathbb F_q$, we also obtain sharp estimates except for two endpoints.
