Let $C_\natural$ be the set of all even continuous functions on the real axis, $E$ be a non-empty set on $(0,+\infty)$, $\mathcal{R} f(x,t)$ be the spherical mean of the function $f\in C_\natural$ with center at the point $x\in E$ and radius $t>0$ with respect to Bessel convolution. The following problems arise for the operator $\mathcal{R}$: 1) find out whether a given set $E$ is an injectivity set of the transform $\mathcal{R}$; 2) if $E$ is not an injectivity set, then characterize all functions $f\in C_\natural$ such that $\mathcal{R} f(x,t)=0$ on $E\times(0,+\infty)$; 3) if $E$ is an injectivity set, then restore $f$ from the values of $\mathcal{R} f(x,t)$ on $E\times(0,+\infty)$. In this paper we obtain a solution of problems 1 and 2 for an arbitrary set $E\subset(0,+\infty)$, as well as a solution of problem 3 for the case when $E$ is a finite set of injectivity. It is shown that functions from the kernel of the transform $\mathcal{R}$ can be described in terms of series of the eigenfunctions of the Bessel operator converging in the space of distributions. It follows, in particular, that the set $E$ is not the injectivity set of the transform $\mathcal{R}$ if and only if it is contained in the set of zeros of some eigenfunction of the Bessel operator. Moreover, if $E=\{r_1,\ldots,r_m\}$ is a finite injectivity set, we find a class of inversion formula for the  transform $\mathcal{R}$ that depend on a set of polynomials $p_1,\ldots,p_m$. It is assumed that  $p_1,\ldots,p_m$ have a sufficiently high degree and satisfy some conditions related to the zeroes of  Fourier – Bessel transform of Dirac measures with supports at points $r_1, \ldots, r_m$.