How is a characteristic subgroup verified? I know the definition of a characteristic subgroup: $\sigma (H)=H$ for all $\sigma \in \text{aut}\, G$ where $H \leq G$. But, I do not understand how $\sigma$ is defined. Surely we can map $H$ to $H' \varsupsetneq H$ in any case? 
For example, let us take $G=C_6$ the cyclic group of order 6, and $H=\langle g^2 \rangle = \{1, g^2, g^4\}$ where $g \in G$. Then define the mapping $\sigma : g^k \mapsto g^{k+1}$. Thus, $\sigma(H)=\{g, g^3, g^5\}\neq H$. I chose this particular group because I read on Wikipedia that subgroups of cyclic groups are characteristic.

Your $\sigma$ is not an automorphism. It does not map the identity element to the identity element. As you say in the definition you only require that $\sigma(H) = H$ for all automorphisms $\sigma$. Not for all bijections or all functions or all homomorphisms.
