= Solution
Partition the vertex set into disjoint sets $A$ and $B$ of sizes $\lfloor n/2\rfloor$ and $\lceil n/2\rceil$. The <induced subgraph> on $A$ has distribution
$$
G(\lfloor n/2\rfloor,p).
$$
For $p=n^{-0.9}$,
$$
p|A|\sim\frac12n^{0.1}\longrightarrow\infty,
$$
so part (b) shows that $A$ contains a triangle with probability tending to one.
On that event, choose a triangle using only the edges internal to $A$. The edges from its three vertices to $B$ remain independent of this choice. The probability that all $3|B|$ such edges are absent is
$$
(1-p)^{3|B|}
\leq e^{-3p|B|}
\longrightarrow0,
$$
because $p|B|\sim n^{0.1}/2$. Thus, with probability tending to one, some vertex of $B$ is adjacent to a vertex of the triangle. Those four vertices and the four required edges form the given graph $H$, a <triangle with an attached leaf in a binomial random graph>. Therefore
$$
\boxed{\mathbb P(F)\longrightarrow1}.
$$
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