I will at least attempt a mathematical formulation of the question. We have a finite set $C$ of candidates, and a finite set $E$ of electors. I will assume that we are just modelling what happens in some kind of secret ballot after campaigning has finished, so each elector makes some kind of private choice, which will lie in some set $F(C)$ depending on $C$. Reasonable symmetry conditions indicate that $F(C)$ should be functorial for bijections of $C$, and independent of the elector. Thus, the collection of choices is a point in $\text{Map}(E,F(C))$. Let $P_n(C)$ denote the set of subsets of size $n$ in $C$. Our task is then to produce a map $\sigma\colon\text{Map}(E,F(C))\to P_n(C)$ with some good properties, yet to be specified. The first obvious property is that it should be natural for bijections of $E$ and $C$. As $E$ does not appear in the codomain of $\sigma$, this means we have a natural map $F(C)^e/\Sigma_e\to P_n(C)$, where $e=|E|$.
Now let $\text{Ord}(C)$ denote the set of total orderings on $C$. This is the same as the set of bijections from the set $c=\{0,1,\dotsc,|C|-1\}$ to $C$. By Yoneda, any point $f\in F(c)$ gives a natural map $f_*\colon\text{Ord}(C)\to F(C)$. The composite $$ \text{Map}(E,\text{Ord}(C)) \xrightarrow{f_{**}} \text{Map}(E,F(C)) \xrightarrow{\sigma} P_n(C) $$ looks like something to which we could apply Gibbard-Satterthwaite.
EDIT: As Tom points out in the comments, there are problems with tie-breaking. Specifically, it could happen that $E=C$, and it could happen that for each $c\in C$ there is a point $\eta(c)\in F(C)$ which expresses the preference "$c$ should get in, and I don't care about anything else". In that case $\sigma(\eta)$ would have to be an $\text{Aut}(C)$-invariant point in $P_n(C)$, which is impossible if $0<n<|C|$. However, one could ask for a map $\sigma$ as above, but only defined on the set $\text{Map}(E,F(C))'$ of points with trivial $\text{Aut}(C)$-isotropy. Unless $E$ is small relative to $C$ or $F$ is unreasonably degenerate, there should be points $f\in F(c)$ such that $f_{**}$ lands in $\text{Map}(E,F(C))'$.