The answer is "NO".
It follows from existence of exotic smooth involutions of sphere.

Say, this movie explains a construction of an involution $\iota$ of $\mathbb S^5$ such that
the quotient $\Pi=\mathbb S^5/\iota$ is homotopy equivalent, but not not homeomorphic to $\mathbb R\mathrm P^5$.

Consider composition
$$\Pi\longrightarrow^{\!\!\!\!\!h}\mathbb R\mathrm P^5\longrightarrow^{\!\!\!\!\!e}\mathbb R\mathrm P^{N-1},$$
where $h$ is the homotopy equivalence and $e$ is the standard embedding.

If $N$ is large, Whitney embedding theorem says that you can perterb this composition into smooth embedding $\Pi\to\mathbb R\mathrm P^{N-1}$.

Passing to the double cover of $\Pi$ and $\mathrm P^{N-1}$,
you get a funny embedding $\mathbb S^5\hookrightarrow \mathbb S^{N-1}\subset \mathbb R^N$,
which gives a counterexample.
(If the image admits is an odd homeomorphism to $\mathbb S^5$ then
$\Pi$ has to be homeomorphic to $\mathbb R\mathrm P^5$.)