  Let $X$ be a finite CW-complex having mod $p$ cohomology isomorphic to a
wedge of three spheres $\mathbb{S}^n\vee \mathbb{S}^m \vee \mathbb{S}^l,~ 1\leq
n \leq m \leq l$. The aim of this paper is to determine the fixed point sets of
actions of the cyclic group of prime order on $X.$ We also classify the orbit
spaces of free actions of $G=\mathbb{Z}_p, p$ a prime or
$G=\mathbb{S}^d,~d=1,3,$ on $X$ and derive the Borsuk-Ulam type results.
