ERA question when error in inning

[Butch7999]

mid50s raised a valid and salient point (9.12.a.5) we'd overlooked, but Jim65 (thank you, sir) beats us to a rebuttal.

9.12
(a) The official scorer shall charge an error against any fielder:
(1) whose misplay (fumble, muff or wild throw)... prolongs the presence on the bases of a runner or permits a runner to advance one or more bases...
(5) whose wild throw permits a runner to reach a base safely, when in the scorer’s judgment a good throw would have put out the runner,
unless such wild throw is made attempting to prevent a stolen base;
(6) whose wild throw in attempting to prevent a runner’s advance permits that runner or any other runner to advance one or more bases
beyond the base such runner would have reached had the throw not been wild;

and

(d) The official scorer shall not charge an error against:
(1) the catcher when the catcher, after receiving the pitch, makes a wild throw attempting to prevent a stolen base,
unless the wild throw permits the stealing runner to advance one or more extra bases or permits any other runner
to advance one or more bases;
(2) any fielder who makes a wild throw if in the scorer’s judgment the runner would not have been put out with ordinary effort
by a good throw, unless such wild throw permits any runner to advance beyond the base he would have reached had the throw not been wild;

So mid50s, you are correct that no error is charged on the overthrow allowing the steal of 2nd -- however, an error is charged
(or should be) on the runner's advance to 3rd as a result of the overthrow.
Because the runner is on 3rd as the result of an error, and not merely safe at 2nd, the run he scores on an ensuing single is unearned.
The run is earned if he scores, as Jim65 posits, on an ensuing home run (or, we'd think, on any extra-base hit).