Let $S$ be the dyadic solenoid.
Let $x\in S$, and let $X$ be the union of all arcs (homeomorphic copies of $[0,1]$) in $S$ containing $x$.
$X$ is called a composant of $S$.
It is well-known that $X$ is a dense first category one-to-one continuous image of the reals, and that $S$ is a homogeneous continuum.
Now let $y$ and $z$ be any two points in $S\setminus X$.
Is there a self-homeomorphism $h:S\to S$ such that $h[X]=X$ and $h(y)=z$?
In this paper of J.Kwapisz I have found the following
Theorem 1. Any homeomorphism $h$ of the dyadic solenoid $S$ is isotopic to the "affine" homeomorphism of the form $g:x\mapsto \pm(2^n x+b)$ for some $n\in\mathbb Z$ and some $b\in S$.
If $h$ preserves the path-connected component $X$ of the neutral element, then so does the affine homeomorphism $g$, which implies that $b\in X$. It follows that for any $y\in S\setminus X$ the image $h(y)$ belongs to $g(y)+X\subset \pm 2^{\mathbb Z}y+X$, which is contained in a countable union of path-connected components and hence cannot be an arbitrary element $z\in S\setminus X$.
So, the answer to the original question is negative.
The same negative answer holds for homeomorphisms of any (not necessarily dyadic) solenoid.
