  A mixed extension of a graph $G$ is a graph $H$ obtained from $G$ by
replacing each vertex of $G$ by a clique or a coclique, whilst two vertices in
$H$ corresponding to distinct vertices $x$ and $y$ of $G$ are adjacent whenever
$x$ and $y$ are adjacent in $G$. If $G$ is the path $P_3$, then $H$ has at most
three adjacency eigenvalues unequal to $0$ and $-1$. Recently, the first author
classified the graphs with the mentioned eigenvalue property. Using this
classification we investigate mixed extension of $P_3$ on being determined by
the adjacency spectrum. We present several cospectral families, and with the
help of a computer we find all graphs on at most $25$ vertices that are
cospectral with a mixed extension of $P_3$.
