Let $G$ be a finite group scheme operating on an algebraic variety $X$, both defined over an algebraically closed field $k$. The paper first investigates the properties of the quotient morphism $X\longrightarrow X/G$ over the open subset of $X$ consisting of points whose stabilizers have maximal index in $G$. Given a $G$-linearized coherent sheaf on $X$, it describes similarly an open subset of $X$ over which the invariants in the sheaf behave nicely in some way. The points in $X$ with linearly reductive stabilizers are characterized in representation theoretic terms. It is shown that the set of such points is nonempty if and only if the field of rational functions $k(X)$ is an injective $G$-module. Applications of these results to the invariants of a restricted Lie algebra ${\frak g}$ operating on the function ring $k[X]$ by derivations are considered in the final section. Furthermore, conditions are found ensuring that the ring $k[X]^{\frak g}$ is generated over the subring of $p$th powers in $k[X]$, where $p={\rm char}\,k>0$, by a given system of invariant functions and is a locally complete intersection.