We prove the formula

for the equivariant category of the wedge $X\vee Y$. As a direct application, we have that the wedge $\bigvee _{i=1}^m X_i$ is $G$-contractible if and only if each $X_i$ is $G$-contractible, for each $i=1,\ldots ,m$. One further application is to compute the equivariant category of the quotient $X/A$, for a $G$-space $X$ and an invariant subset $A$ such that the inclusion $A\hookrightarrow X$ is $G$-homotopic to a constant map $\overline {x_0}\,:\,A\to X$, for some $x_0\in X^G$. Additionally, we discuss the equivariant and invariant topological complexities for wedges. For instance, as applications of our results, we obtain the following equalities:

\begin{align*} \text{TC}_G(X\vee Y)&=\max \{\text{TC}_G(X),\text{TC}_G(Y),\text{cat}_G(X\times Y)\},\\ \text{TC}^G(X\vee Y)&=\max \{\text{TC}^G(X),\text{TC}^G(Y),_{X\vee Y}\text{cat}_{G\times G}(X\times Y)\}, \end{align*}

for $G$-connected $G$-CW-complexes $X$ and $Y$ under certain conditions.