  In this paper, we calculate the $n+3$, $n+4$ dimensional homotopy groups of
indecomposable $\mathbf{A}_n^2$-complexes after localization at 2. This job is
seen as a sequel to P.J. Hilton's work on the $n+1,n+2$ dimensional homotopy
groups of $\mathbf{A}_n^2$-complexes (1950-1951). The main technique used is
analysing the homotopy property of $J(X,A)$, defined by B. Gray for a CW-pair
$(X,A)$, which is homotopy equivalent to the homotopy fibre of the pinch map
$X\cup CA\rightarrow \Sigma A$. By the way, the results of these homotopy
groups have been used to make progress on recent popular topic about the
homotopy decomposition of the (multiple) suspension of oriented closed
manifolds.
