The notion of the "fractional charge" carried by anyons in an SET phases is actually subtly different from the projective representations that appear at the boundary of a one-dimensional SPT. To illustrate this distinction, consder the case of a $U(1)$ symmetry. One can attempt to define a "projective representation" corresponding to, say, 1/3 charge as observed in the quasiparticles of the fractional quantum Hall effect:
$$ V(e^{i\theta}) = e^{i\theta/3}$$
Observe, however, that this "projective representation" differs from the trivial representation only be a phase factor. Given that the action of a symmetry on the boundary of an SPT is only determined up to a phase factor, there is no well-defined sense in which an SPT boundary can be said to carry fractional $U(1)$ charge.
The reason why anyons in an SET are different and it is meaningful for them to carry fractional $U(1)$ charge is a rather subtle. We have to go back and understand why, exactly, the symmetry action on the boundary of an SPT is only defined up to a phase factor. Basically, one consider a one-dimensional chain with open boundary conditions. The system acts linearly (non-projectively) on the whole chain. By reducing to the low-energy modes at the boundary, we conclude that the system still acts linearly on the boundary (which comprises two points $a$ and $b$). Locality implies that we can decompose the symmetry action on the boundary as $U(g) = V_a(g) \otimes V_b(g)$, where $V_a(g)$ acts on the left edge and $V_b(g)$ acts on the right edge. But this decomposition is not quite unique, since we can multiply $V_a(g)$ by any phase factor $\beta(g)$ so long as we also multiply $V_b(g)$ by $\beta(g)^{-1}$. Hence the phase factor ambiguity.
It might seem that the situation for anyons is similar. Consider a state $|a,\overline{a}\rangle$ containing an anyon $a$ and its antiparticle $\overline{a}$, well separated from each other. Then we can decompose the symmetry action on this state as a product $V_a(g) \otimes V_{\overline{a}}(g)$. By a similar argument as before, it would seem that $V_a(g)$ is only defined up to a phase factor transformation.
But, there is something else we can do! Suppose that $n$ copies of $a$ fuse to the vacuum. (For simplicitly, I'll assume $n=3$). This allows us to consider the state $|a,a,a\rangle$ and decompose the symmetry action on this state as $U(g) = V_a(g) \otimes V_a(g) \otimes V_a(g)$. Now observe that the only phase factor transformations we can do on $V_a(g)$ are of the form $V_a(g) \to \beta(g) V_a(g)$, where $\beta(g)^3 = 1$. This does not allow us to eliminate the charge 1/3 projective representation described above, hence charge 1/3 is actually distinct from charge 0.
The upshot is that, whereas projective representation of an SPT boundary is classified by the cohomology group $H^2(G, U(1))$, for an anyon of which $n$ copies fuse to the vacuum it is actually $H^2(G, Z_n)$ instead.
There are also more general things that can happen in an SET, for example the symmetry can change an anyon into a different anyon type.
