Let $\Pi$ be a cuspidal automorphic representation for GL(3) over a number field. We fix an integer $k \geq 2$ and we assume that the symmetric $m$th power lifts of $\Pi$ are automorphic for $m \leq k$, cuspidal for $m < k$, and that certain associated Rankin–Selberg products are automorphic. In this setting, we bound the number of cuspidal isobaric summands in the $k$th symmetric power lift. In particular, we show it is bounded above by 3 for $k \geq 7$, and bounded above by 2 when $k \geq 19$ with $k$ congruent to 1 mod 3. We will also discuss the analogous problem for GL(4).