Let,
$1)$ Either Kevin or Heather, or both, are chatting : $K \;\vee\; H$
$2)$ Either Randy or Vijay, but not both, are chatting : $R\oplus V$
$3)$ If Abby is chatting, so is Randy : $A \rightarrow R$
$4)$ Vijay and Kevin are either both chatting or neither is : $(V \wedge K) \vee (\sim V \wedge \sim K)$ i.e. $V \Leftrightarrow K$
$5)$ If Heather is chatting, then so are Abby and Kevin : $H \rightarrow A\;\wedge \;K$
Now, we have to find the assignment for $K,H,R,V$ and $A$ so that whole system will be consistent means some assignment of truth values for these variables so that all the statements must be true.
Now, Let's start with $5)$ $H \rightarrow A\;\wedge \;K$ and $1)$ $K \;\vee\; H,$
For this, Possible assignments of truth values($T$ = True and $F$ = False) are :-
$1) A= F , K = T , H = F$
$2) A= T , K = T , H = F$
$3) A= T , K = T , H = T$
Now, statement $3)$ $A \rightarrow R$ will be true when :-
$1) A= F , K = T , H = F , R = T$
$2) A= F , K = T , H = F , R = F$
$3) A= T , K = T , H = F , R = T$
$4) A= T , K = T , H = T , R = T$
Now, statement $2)$ $R\oplus V$ will be true when either $R=T, V = F$ (or) $V=T , R = F$ , So, possible assignments will be :-
$1) A= F , K = T , H = F , R = T , V = F$
$2) A= F , K = T , H = F , R = F , V = T$
$3) A= T , K = T , H = F , R = T , V = F$
$4) A= T , K = T , H = T , R = T , V = F$
Now, last statement $4)$ $(V \wedge K) \vee (\sim V \wedge \sim K)$ i.e. $V \Leftrightarrow K$ will be true when both $K$ and $V$ are True or False.
So, in above assignments $1), 3)$ and $4)$ are not possible. Only possibility is assignment $2)$ i.e. $A= F , K = T , H = F , R = F , V = T.$ It means only Kevin and Vijay are chatting.
So, Answer is :- Only Kevin and Vijay are chatting.