Representation theory of finite groups

The alternating group $A_4$ (order $12$) has $4$ conjugacy classes and exactly $3$ one-dimensional complex representations. What is the degree of its remaining irreducible representation?

Which statements are true for complex representations of a finite group $G$? (Choose all that apply.)

A representation of $S_3$ has character $\chi = (4, 0, 1)$ on the classes of the identity, the transpositions and the $3$-cycles (sizes $1, 3, 2$). Decompose it into irreducibles ($\mathbf{1}$ trivial, $\varepsilon$ sign, $V$ two-dimensional).

$S_4$ permutes the $6$ two-element subsets of $\{1, 2, 3, 4\}$. The character of this permutation representation on the classes $1, (12)(34), (12), (123), (1234)$ (sizes $1, 3, 6, 8, 6$) is $(6, 2, 2, 0, 0)$. Compute $\langle \chi, \chi \rangle$, the number of irreducible constituents counted with squared multiplicity.

In the proof of Maschke's theorem we replace a projection $\pi : V \to W$ by $\pi_0 = \frac{1}{|G|}\sum_g g\pi g^{-1}$. Which step uses that the characteristic of $k$ does not divide $|G|$?

Which of these representations are not completely reducible? (Choose all that apply.)

Let $\omega$ be a non-trivial one-dimensional character of $A_3 = \{1, (123), (132)\} \le S_3$. What is $\operatorname{Ind}_{A_3}^{S_3} \omega$?

Which Lean statement says that the shear representation over $\mathbb{F}_2$ is not completely reducible, as in maschke_fails? (Here L abbreviates Submodule.span (ZMod 2) {((1 : ZMod 2), (0 : ZMod 2))}.)

Write multiplicities(chi, table, sizes): given a real character chi (values on the conjugacy classes), the character table (a list of rows of real irreducible characters, same class order) and the class sizes, return the list of multiplicities $m_i = \frac{1}{|G|}\sum_C |C|\,\chi(C)\,\chi_i(C)$ as integers, where $|G| = \sum_C |C|$.