Semimartingales of processes with independent increments and enlargement of filtration

Let $X$ be a process with independent increments, $\mathcal{F} = (\mathcal{F}_t )$, $0 \le t \le T, \mathcal{F} = \sigma (X_s ,s \le t)$ a natural filtration. Denote
$$ G_t = \sigma \{ {X_s ,s \le t; X^c ( T ); p\{ ] {0;T} ]; A \in \mathcal{B} \}} \},\qquad t \le T, $$
where ${X^c }$ is a continuous martingale component, ${p\{ { ] {0;T} ]; A \in \mathcal{B}}\}}$ is the integer-valued Poisson measure generated by ${X,\mathcal{B}}$ is the Borel $\sigma $-algebra. The paper discusses conditions under which any process $Y$ being a semimartingale with respect to filtration $F$ is also a semimartingale with respect to filtration $G$.