We denote for an integer $n>1$ its square-free kernel as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p,$$ with the definition $\operatorname{rad}(1)=1$. You can see this definition and the properties of this arithmetic function for example from this Wikipedia. In this post also we denote the Euler's totient function as $\varphi(n)$.
Today I am trying to explore claims involving the product of distinct primes dividing $n$, that is $\operatorname{rad}(n)$ and number theoretic functions. In this post we study the case about the Euler's totient function.
Claim 1. It is easy to prove that if $n$ has the form $n=2^{\alpha}5^{\beta}$, where $\alpha\geq 1$ and $\beta\geq 1$ are integers, then our integer $n$ satisfies $$\operatorname{rad}(2n+\varphi(n))=\operatorname{rad}(n+2\varphi(n)).\tag{1}$$
Claim 2. It is easy to prove that if $n$ has the form $n=2^{\alpha}3^{\beta}$, where $\alpha\geq 1$ and $\beta\geq 1$ are integers, then our integer $n$ satisfies $$\operatorname{rad}(n-\varphi(n))=\operatorname{rad}(n+\varphi(n)).\tag{2}$$
Question.
A) The first solution $m$ of $(1)$ that has a prime factor $P\notin\{2,5\}$ is $m=22110=2\cdot3\cdot 5\cdot 11\cdot 67$. Are there infinitelty many solutions $m$ of $(1)$ such that $m\notin$A033846 from the OEIS (these are integers of the form $n=2^{\alpha}5^{\beta}$, for some integers $\alpha\geq 1$ and $\beta\geq 1$)?
B) Prove or refute that if $n$ is an integer satisfying $(2)$ then $n$ has the form $n=2^{\alpha}3^{\beta}$, where $\alpha\geq 1$ and $\beta\geq 1$ are integers (that is the sequence A033845 from the OEIS).
Many thanks.
If isn't possible a full answer add what work can be done, I say it because I think that this kind of questions are very difficult to solve. Also I would like to add that I don't know if the equations $(1)$ or $(2)$ are in the literature. This is the link for OEIS.