  We consider the Moore-Gibson-Thompson equation with memory of type II $$
\partial_{ttt} u(t) + \alpha \partial_{tt} u(t) + \beta A \partial_t u(t) +
\gamma Au(t)-\int_0^t g(t-s) A \partial_t u(s){\rm d} s=0 $$ where $A$ is a
strictly positive selfadjoint linear operator (bounded or unbounded) and
$\alpha,\beta,\gamma>0$ satisfy the relation $\gamma\leq\alpha\beta$. First, we
prove a well-posedness result without requiring any restriction on the total
mass $\varrho$ of $g$. Then we show that it is always possible to find memory
kernels $g$, complying with the usual mass restriction $\varrho<\beta$, such
that the equation admits solutions with energy growing exponentially fast. In
particular, this provides the answer to a question raised in "F. Dell'Oro, I.
Lasiecka, V. Pata, J. Differential Equations 261 (2016), 4188-4222".
