Question exactly as given on past exam:
Consider the group $G=(\mathbb{Z}/15\mathbb{Z})^\times$ (under multiplication). Let $H$ be the subgroup generated by $2$ (that is, $H = \langle 2 \rangle$). Write out the elements of $G,H$ and $G$/$H$. State $|G|,|H|$ and $[G:H]$
Now I want to list the elements of the following(With attempt underneath)
$$G = \{1_{15},2_{15},4_{15},7_{15},8_{15},11_{15},13_{15},14_{15}\}$$
With H, I don't know if it means that it is a subgroup of $G$ or of the integers. I will do my attempt at both $$H_1 = 2_{15},4_{15},8_{15},1_{15}\;, \; H_2 = 2,4,8,16,32,...,2^n$$ Now based on the following question, $H_1$ makes more sense. $$G/H = 7_{15},11_{15},13_{15},14_{15}$$ I think it is asking for cardinality $$|G| = 8$$ $$|H| = 4$$
I don't even know what this one is:
$[G:H] = 4$
I agree that $H$ should probably be a subgroup of $G$ rather than the integers, since the problem appears to be framed in terms of $G$. Your explicit sets for $G$ and $H$ look good, and the cardinalities are correct.
As you noted in the comments, you had some misunderstanding with the $/$ and $\setminus$ symbols. Here, $G/H$ refers to the set of cosets of $H$ in $G$. Therefore, each element of $G/H$ is a subset of $G$ with size $|H|$. Also, the notation $[G:H]$ is used to mean $|G/H|$, which is often more easily calculated as $|G|\,/\,|H|$; in this case it is $2$.
All that is left to do is to write $G/H$ down explicitly; have you seen this before?