A triple system of order $v \geq 3$ and index $\lambda$ is faithfully enclosed in a triple system of order $w \geq v$ and index $\mu \geq \lambda$ when the triples induced on some $v$ elements of the triple system of order $w$ are precisely those from the triple system of order $v$. When $\lambda = \mu$, faithful enclosing is embedding; when $\lambda = 0$, faithful enclosing asks for an independent set of size $v$ in a triple system of order $w$. When $\mu = 2 \lambda$, we prove that a faithful enclosing of a triple system of order $v$ and index $\lambda$ into a triple system of order $w$ and index $\mu$ exists if and only if $w \geq \lceil \frac3v-12 \rceil$, $\mu \equiv 0 \pmod \gcd(w-2,6) $, and $(v,w) \not\in (3,5), (5,7) \$.