A threshold function with the Shannon effect for Boolean functions with respect to a symmetric group

We prove that if $A$ is the property of a Boolean function $f(x_1,\dots,x_n)$ of weight $r$ consisting in the fact that $f$ has a trivial inertia group with respect to the group $S_n$, then the function $A(n)=\ln n$ is a threshold function for property $A$.