Is a representation sphere dualizable inside naive G-spectra? Let $G$ be a finite group and let $G$ act on a vector space $V$.  Let $S(V)$ be the representation sphere.  In the monoidal category of $RO(G)$-graded spectra $S(V)$ is invertible, so it's dualizable too.
Is $S(V)$ dualizable in the monoidal category of naive $G$-spectra?
 A: Let $G$ be of order $2$, and let $V$ be the nontrivial one-dimensional representation.  I'll write $S(V)$ for the unit sphere in $V$ (which is just $G$); the one-point compactification of $V$ is then the unreduced suspension of $G$, which I'll call $S^V$.  We can only form spectra from $G$-spaces with fixed basepoint, so we need to consider $S(V)_+=G_+$ rather than $S(V)$ itself.  There is a cofibration $G_+\to S^0\to S^V$, and $S^0$ is its own dual, so the question for $S^V$ is equivalent to the question for $G_+$.
In the category of genuine $G$-spectra, $G_+$ is its own dual, which means that there are unit and counit maps $S^0\xrightarrow{\eta}G_+\wedge G_+\xrightarrow{\epsilon}S^0$ making certain diagrams commute.  In the naive category, $\{S^0,G_+\wedge G_+\}^G$ is the colimit of the unstable mapping sets $[S^n,S^n\wedge G_+\wedge G_+]^G$, which are trivial because $S^n\wedge G_+\wedge G_+$ is free away from the basepoint.  Thus, we cannot construct the required map $\eta$, and $G_+$ is not self-dual.  The same applies if we replace one of the $G_+$'s by something else, so $G_+$ is not dualisable.
A: There is a strongly related result of Gaunce Lewis that I would like to advertise.   We have different kinds of $G$-spectra for different ``universes'' $U$, which are countable sums of representations that include $\mathbf{R}$ and countably many copies of each representation they contain.  Naive $G$-spectra are indexed on the trivial universe, genuine ones are indexed on the complete universe, which contains all irreducible representations.  Gaunce proved that the $U$-suspension $G$-spectrum of an orbit $G/H_+$ (I've added a disjoint basepoint) is dualizable if and only if $G/H$ embeds as a sub $G$-space of a representation in the universe $U$. Neil's example is the simplest illustration of this general result. (Tyler, no problem with cofibrancy here: Neil is working in the Spanier-Whitehead category; Gaunce is working with Lewis-May $G$-spectra.)
