In Stein's online textbook here, in the section "Explicit Basis of the Eisenstein Subspace", he defines the Eisenstein series
$$E_{k,\chi,\psi}(q)=c_0+\sum_{n=1}^{\infty}\left(\sum_{d|n}\psi(d)\chi(n/d)d^{k-1}\right)q^n$$
It is then stated that if $t$ is a positive integer and $k$ is a natural such that $\psi(-1)\chi(-1)=(-1)^k$ then other than in some exceptional cases $E_{k,\chi,\psi}\left(q^t\right)$ defines an element of $M_k(\Gamma(RLt))$ where $R$ and $L$ are the conductors of $\psi$ and $\chi$ respectively.
I am confused for a couple of reasons
If a function is a modular form in $\Gamma(N)$, then it is periodic mod $N$. This means that it should have a $q$ expansion where $q=e^{2\pi i z/N}$. For this reason, raising $q$ to the power of $t$ should decrease the period, not increase it, i.e $E_{k,\chi,\psi}(q^t)$ should be an element of $M_k(\Gamma(RL/t))$ and $E_{k,\chi,\psi}(q^{1/t})$ should be an element of $M_k(\Gamma(RLt))$. Not what is said.
Am I correct in assuming that when the modular form is being defined, $q=e^{2\pi i z/RL}$? This is what it should need to be for the periods to match up if I am not mistaken.
Stein defines the Eisenstein subspace only for the group $\Gamma_1(N)$. There is indeed a typo, it should be $\Gamma_1(RLt)$ instead of $\Gamma(RLt)$. Moreover $q$ denotes $e^{2\pi iz}$ as usual. Also, in general, if $f(q)$ is a modular form on $\Gamma_1(N)$ then $f(q^t)$ is a modular form on $\Gamma_1(Nt)$. These things are also explained in the book A first course on modular forms by Diamond and Shurman.
Here is an example (pointed out by the OP). Take $k \geq 2$ even, $\chi$ is the unique character modulo $2$, and $\psi$ is the trivial character. Then as explained in Stein's textbook, $E_{k,\chi,\psi}$ belongs to $M_k(\Gamma_1(2))$.
What about $E_{k,\chi,\psi}(q^{1/2})$? One can show as an exercise that if $f(z)$ is a modular form on $\Gamma_1(N)$, then $f(q^{1/d})$ is a modular form on $\Gamma(N)$ for every $d$ dividing $N$ (here $q^{1/d} = e^{2\pi iz/d}$). Thus, in your case, $E_{k,\chi,\psi}(q^{1/2})$ is a modular form on $\Gamma(2)$.