Google Inc's free Android smartphone software, already a big hit with consumers, is starting to win the hearts of investors.

The world's No. 1 Internet search company offered a peek at its mobile business during quarterly results on Thursday, revealing that the business was generating revenue at an annual run rate of $2.5 billion, up from $1 billion last year.

That helped Google sail past third-quarter financial targets set by analysts, sending its shares up nearly 6 percent to $591.68 on Friday and easing some of Wall Street's concerns that mobile returns might not justify the investment.

People just haven't given them any credit for that division. I think it could be a huge part of the overall company, said Pat Adams, portfolio manager at the Dunham Loss Averse Growth Fund, which owns Google shares.

There are so many more mobile devices out there than there are PCs, Adams said. What they did was brilliant to give that operating system away to get the search part of it, he added.

Google lets phone makers such as Samsung Electronics, LG Electronics and HTC Corp use its Android software for free, banking on consumers using those phones to visit Google's advertising-supported website to search for information.

The booming popularity of smartphones has frustrated many of the established giants of the computer industry, from Microsoft Corp to Hewlett-Packard Co.

For Google, whose business is built upon people using its search engine, making the transition from the personal computers to mobile devices is crucial.

The company has stepped up investments in its mobile business, which competes with iPhone-maker Apple Inc. Google's Android mobile software -- already the world's most-used smartphone platform -- powers 190 million devices, up from 135 million in mid-July.

The explosion of Android devices, as well as the availability of Google search on Apple's iPhones, has made Google even more dominant in mobile search than on the desktop PC, according to JP Morgan analyst Doug Anmuth who pegged Google's mobile search market share at 90 percent.

That strong position accounts for the sharp, 28 percent uptick in paid clicks on search ads that Google experienced during the third quarter, Anmuth said in a note to investors.

Those ads appear to command lower rates than PC search ads, analysts noted. But some analysts said they expect that to change over time, especially as Google creates new forms of advertising that take advantage of a user's location.

Mobile advertising sales is but one component of what analysts believe could be a broader wireless opportunity for Google. The company has begun offering coupon deals, and could make money through retailer loyalty programs and its recently launched Google Wallet, a free service which allows shoppers to use their mobile phones to pay for purchases.

MOTOROLA BET

More concerning for some investors is Google's plan to acquire mobile phone maker Motorola Mobility Holdings for $12.5 billion.

The deal will give Google access to one of the largest patent libraries in the wireless industry, as well as hardware manufacturing operations that will allow it to develop its own line of smartphones.

But some worry that Google is entering a low-margin hardware business in which it has no experience, and that the move could jeopardize its relationships with other phone makers that use Android.

BGC Partners analyst Colin Gillis said he did not think Google's increase in mobile ad revenue would make investors feel any better about the Motorola deal, which is expected to close this year or early in 2012.

You could argue the Motorola deal puts some of that revenue at risk, he said, noting that some current Android phone makers might see Google as a competitor once it acquires Motorola and reduce their support for Google products. Google has said it plans to operate Motorola as a separate business.

Gillis also noted that the $2.5 billion annual run rate in Google's mobile business, while impressive, remains less than 10 percent of the company's overall revenue.

And he added that Google may not necessarily have based the $2.5 billion run rate on one quarter's worth of revenue, which would have suggested that Google made $625 million in mobile revenue in the third quarter.

They probably took the last month and multiplied it by 12. It could be the last day, he noted. We have no idea what that number really is.

Whatever the number though, Google's mobile revenue is clearly growing quickly, and for many on Wall Street, that's good enough for now. Many brokerages raised their price targets on Google on Friday, some by as much as 10 percent.

We think mobile is near a massive volume inflection point, wrote Susquehanna Financial Group analyst Herman Leung in a note to investors on Friday.

At these growth rates, we think mobile revenue could be larger than display (advertising revenue) by 2012.

(Reporting by Alexei Oreskovic; Additional reporting by Sayantani Ghosh, Tenzin Pema and Rachana Khanzode in Bangalore; Editing by Richard Chang)