Let $G$ be a finite group and $\mathbb{F}_q$ be a finite field, where $q$ is a prime power. Let $\mathbb{F}_qG$ be a finite semisimple group algebra.
Then Wedderburn decomposition theorem implies that $\mathbb{F}_qG$ is direct sum of matrix rings over division rings, i.e.,$$\mathbb{F}_qG\cong \oplus_{i=1}^t M(n_i, \mathbb{F}_{q_i}).$$
Here $\mathbb{F}_{q_i}$ is a finite extension of $\mathbb{F}_{q}$, $n_i\geq 1$ and $t\geq 1$ are positive integers.
Given: Suppose that $\mathbb{F}_{q}$ is such that $$\mathbb{F}_qG\cong \oplus_{i=1}^t M(n_i, \mathbb{F}_{q}).$$ This means $q_i=q$ for each $i$.
Let $q'\neq q$ be another prime power such that the group algebra $\mathbb{F}_{q'}G$ is again semisimple.
Suppose that $M\left(n, \mathbb{F}_{(q')^r}\right)$ is a Wedderburn component of the group algebra $\mathbb{F}_{q'}G$ for some $r\geq 2$ and any positive integer $n$.
Then I want to show that $M\left(n, \mathbb{F}_{q}\right)$ must be a Wedderburn component of the group algebra $\mathbb{F}_qG$.
I have tried various examples and found this result true. Please help me to prove this.
First of all, I just reminded you that $\mathbb{F}_qG$ is semisimple if and only if $p$ and $|G|$ are co-prime.
Now for the question. The idea is to tensor with $\overline{\mathbb{F}_q}$. It's easy to see that $$\overline{\mathbb{F}_q}G =\mathbb{F}_qG\otimes_{\mathbb{F}_q}\overline{\mathbb{F}_q}\cong \bigoplus M(n_i,\overline{\mathbb{F}_q})$$ On the other hand, we get $$\overline{\mathbb{F}_{q}}G =\mathbb{F}_{q'}G\otimes_{\mathbb{F}_{q'}}\overline{\mathbb{F}_q}\cong \bigoplus M(m_i,\mathbb{F}_{(q')^{r_i}}\otimes_{\mathbb{F}_{q'}}\overline{\mathbb{F}_q})$$ Now using that $\mathbb{F}_{(q')^{r_i}}\otimes_{\mathbb{F}_{q'}}\overline{\mathbb{F}_q}\cong \overline{\mathbb{F}_q}^{r_i} $, we can get $$M(m_i,\mathbb{F}_{(q')^{r_i}}\otimes_{\mathbb{F}_{q'}}\overline{\mathbb{F}_q})\cong M(m_i,\overline{\mathbb{F}_q})^{r_i}$$ From the uniqueness of the decomposition, we get that every $m_i$ is a some $n_i$.
For the claim about the tensor product of the fields look at étale algebres or this post.