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Let $(X,G,\Phi)$ be a dynamical system, where $X$ is compact Hausdorff space,and $G$ is a countable discrete group. We investigate shadowing property andmixing between subshifts and general dynamical systems. For the shadowingproperty, fix some finite subset $S\subset G$. We prove that if $X$ is totallydisconnected, then $\Phi$ has $S$-shadowing property if and only if$(X,G,\Phi)$ is conjugate to an inverse limit of a sequence of shifts of finitetype which satisfies Mittag-Leffler condition. Also, suppose that $X$ is metricspace (may be not totally disconnected), we prove that if $\Phi$ has$S$-shadowing property, then $(X,G,\Phi)$ is a factor of an inverse limit of asequence of shifts of finite type by a factor map which almost liftspseudo-orbit for $S$.  On the other hand, let property $P$ be one of the following property:transitivity, minimal, totally transitivity, weakly mixing, mixing, andspecification property. We prove that if $X$ is totally disconnected, then$\Phi$ has property $P$ if and only if $(X,G,\Phi)$ is conjugate to an inverselimit of an inverse system that consists of subshifts with property $P$ whichsatisfies Mittag-Leffler condition. Also, for the case of metric space (may benot totally disconnected), if property $P$ is not minimal or specificationproperty, we prove that $\Phi$ has property $P$ if and only if $(X,G,\Phi)$ isa factor of an inverse limit of a sequence of subshifts with property $P$ whichsatisfies Mittag-Leffler condition.

Shadowing and mixing on systems of countable group actions